Nisilsin

Of course, when it comes to things like this, it's rarely simple.
Generally speaking, border guards at exit points, especially in airports, do not care if you have permission to enter the country you are going to, only that you were in their country legally. Verifying your eligibility to enter the destination country is the job of the airline agents. When travelling with two passports, one has to take care to show the correct passport to the correct authorities. Generally, when dealing with border guards - as opposed to airline employees - you show them the passport that allows you to be in their country. So it really better be the same one whether you're coming or leaving, otherwise they will rightfully get confused.
The other nuance is that when you check in to your flight they ask for your passport info. This is where you may end up using different passports on different legs of your journey. For example, suppose you are flying New York to Kyiv and have both an American and a Ukrainian passports.
You check in at the airline counter in the New York airport with a Ukrainian passport. You show the customs officer at the security checkpoint your American passport. You board the plane and show your Ukrainian passport. You get off the plane and clear customs in Kyiv with your Ukrainian passport. When you are ready to leave, you check in at the Kyiv airport with your American passport. You show the customs officer your Ukrainian passport. You board the plane and show your American passport. You land and clear customs with your American passport. Now, there is some flexibility here in terms of showing the airline crew in New York either your American or Ukrainian passport because both allow visa free entry into Ukraine. However, if you check in using a different passport than what you eventually show border guards at the destination you may have slight delays because the passenger manifest data provided by the airline to the government will have you listed with a different passport. It ranks pretty low in terms of things that can cause serious trouble, but it's still a bit of a hassle.
I know it's not quite what you asked, but I'm just putting it out there for anyone who decides to go ahead and get a second passport. Stay away from preventable issues.
It might be different for land borders, but I don't have much experience with those.
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As for restrictions and different treatments for citizens vs non-citizens, that's definitely a concern, and it's worth pointing out that having a passport is one thing, but being a citizen is another. I did not magically obtain my Ukrainian citizenship upon applying for a passport - I've had it all along since August 24 1991 and never seized being a Ukrainian citizen. A passport just acts as proof that one is indeed a citizen. The millions of Americans who never bother applying for an American passport are still American citizens.
The other important consideration is residency. For many special kinds of properties and financial instruments, it can make an even bigger difference than citizenship. If you plan to engage in those kinds of activities, you have to careful consider what laws apply to those activity in all the countries where you are a citizen and all countries where you are a resident (depending on conditions, you may be considered a resident of more than one country at a time, you don't even have to be a citizen of any of them).
So yes, once you get deeper into this, it's not exactly simple and clear cut, and you may want to carefully consider what is the best approach for your situation.
But as for the basic question of, can you get a Ukrainian passport if you are living abroad and obtained another citizenship since leaving Ukraine? Yes, it's possible and it's actually rather uncomplicated.

Just came in to Ukraine again with my US passport. When the officer saw my previous 3 pairs of entry/exit stamps within the year, he did a double take. This is the first time I have ever seen a Ukraine immigration officer show any emotion at all. He then shrugged, put another stamp in and told me have a nice day. This is also the first time I have ever heard a Ukraine immigration officer speak.

Of course, when it comes to things like this, it's rarely simple.
Generally speaking, border guards at exit points, especially in airports, do not care if you have permission to enter the country you are going to, only that you were in their country legally. Verifying your eligibility to enter the destination country is the job of the airline agents. When travelling with two passports, one has to take care to show the correct passport to the correct authorities. Generally, when dealing with border guards - as opposed to airline employees - you show them the passport that allows you to be in their country. So it really better be the same one whether you're coming or leaving, otherwise they will rightfully get confused.
The other nuance is that when you check in to your flight they ask for your passport info. This is where you may end up using different passports on different legs of your journey. For example, suppose you are flying New York to Kyiv and have both an American and a Ukrainian passports.
You check in at the airline counter in the New York airport with a Ukrainian passport. You show the customs officer at the security checkpoint your American passport. You board the plane and show your Ukrainian passport. You get off the plane and clear customs in Kyiv with your Ukrainian passport. When you are ready to leave, you check in at the Kyiv airport with your American passport. You show the customs officer your Ukrainian passport. You board the plane and show your American passport. You land and clear customs with your American passport. Now, there is some flexibility here in terms of showing the airline crew in New York either your American or Ukrainian passport because both allow visa free entry into Ukraine. However, if you check in using a different passport than what you eventually show border guards at the destination you may have slight delays because the passenger manifest data provided by the airline to the government will have you listed with a different passport. It ranks pretty low in terms of things that can cause serious trouble, but it's still a bit of a hassle.
I know it's not quite what you asked, but I'm just putting it out there for anyone who decides to go ahead and get a second passport. Stay away from preventable issues.
It might be different for land borders, but I don't have much experience with those.
...
As for restrictions and different treatments for citizens vs non-citizens, that's definitely a concern, and it's worth pointing out that having a passport is one thing, but being a citizen is another. I did not magically obtain my Ukrainian citizenship upon applying for a passport - I've had it all along since August 24 1991 and never seized being a Ukrainian citizen. A passport just acts as proof that one is indeed a citizen. The millions of Americans who never bother applying for an American passport are still American citizens.
The other important consideration is residency. For many special kinds of properties and financial instruments, it can make an even bigger difference than citizenship. If you plan to engage in those kinds of activities, you have to careful consider what laws apply to those activity in all the countries where you are a citizen and all countries where you are a resident (depending on conditions, you may be considered a resident of more than one country at a time, you don't even have to be a citizen of any of them).
So yes, once you get deeper into this, it's not exactly simple and clear cut, and you may want to carefully consider what is the best approach for your situation.
But as for the basic question of, can you get a Ukrainian passport if you are living abroad and obtained another citizenship since leaving Ukraine? Yes, it's possible and it's actually rather uncomplicated.

Of course, when it comes to things like this, it's rarely simple.
Generally speaking, border guards at exit points, especially in airports, do not care if you have permission to enter the country you are going to, only that you were in their country legally. Verifying your eligibility to enter the destination country is the job of the airline agents. When travelling with two passports, one has to take care to show the correct passport to the correct authorities. Generally, when dealing with border guards - as opposed to airline employees - you show them the passport that allows you to be in their country. So it really better be the same one whether you're coming or leaving, otherwise they will rightfully get confused.
The other nuance is that when you check in to your flight they ask for your passport info. This is where you may end up using different passports on different legs of your journey. For example, suppose you are flying New York to Kyiv and have both an American and a Ukrainian passports.
You check in at the airline counter in the New York airport with a Ukrainian passport. You show the customs officer at the security checkpoint your American passport. You board the plane and show your Ukrainian passport. You get off the plane and clear customs in Kyiv with your Ukrainian passport. When you are ready to leave, you check in at the Kyiv airport with your American passport. You show the customs officer your Ukrainian passport. You board the plane and show your American passport. You land and clear customs with your American passport. Now, there is some flexibility here in terms of showing the airline crew in New York either your American or Ukrainian passport because both allow visa free entry into Ukraine. However, if you check in using a different passport than what you eventually show border guards at the destination you may have slight delays because the passenger manifest data provided by the airline to the government will have you listed with a different passport. It ranks pretty low in terms of things that can cause serious trouble, but it's still a bit of a hassle.
I know it's not quite what you asked, but I'm just putting it out there for anyone who decides to go ahead and get a second passport. Stay away from preventable issues.
It might be different for land borders, but I don't have much experience with those.
...
As for restrictions and different treatments for citizens vs non-citizens, that's definitely a concern, and it's worth pointing out that having a passport is one thing, but being a citizen is another. I did not magically obtain my Ukrainian citizenship upon applying for a passport - I've had it all along since August 24 1991 and never seized being a Ukrainian citizen. A passport just acts as proof that one is indeed a citizen. The millions of Americans who never bother applying for an American passport are still American citizens.
The other important consideration is residency. For many special kinds of properties and financial instruments, it can make an even bigger difference than citizenship. If you plan to engage in those kinds of activities, you have to careful consider what laws apply to those activity in all the countries where you are a citizen and all countries where you are a resident (depending on conditions, you may be considered a resident of more than one country at a time, you don't even have to be a citizen of any of them).
So yes, once you get deeper into this, it's not exactly simple and clear cut, and you may want to carefully consider what is the best approach for your situation.
But as for the basic question of, can you get a Ukrainian passport if you are living abroad and obtained another citizenship since leaving Ukraine? Yes, it's possible and it's actually rather uncomplicated.

The basic test for Ukrainian citizenship in this case is that he was living in Ukraine at the time of the declaration of independence, in August 1991. Since he left in '98, this should be easy peasy.
The first step is to visit the nearest consulate or embassy and ask them about getting the passport. I'm from Canada so the minor details may differ, but the overall process should be the same in the USA.
Since there are no currently valid documents aside from birth certificate (which is not sufficient for citizenship, but bring it along anyway), they will ask you to fill out a form which they will use to try and find the records in Ukraine's internal registry. The information needed for this form is the last known address where he lived in Ukraine. So if he was 12 and does not remember exactly, now is the time to check any old paperwork or ask parents or what not and find out the exact address. Together with the name, dob and other details (which should be easy to provide) they will use it to match up the info and send a confirmation to your embassy that yes, so and so is a Ukrainian citizen and eligible for passport.
This part takes anywhere from a couple of weeks to a couple of months, and costs nothing! There is no way to speed it up, it just depends on how lucky you are and how quickly they find the records back in Ukraine.
The next part is to actually apply for passport. Check with your consulate for the exact details, such as amount and form of payment (mine was a tad bit over $100, payable as a bank draft) and if you need to bring ready made photos. Mine did not accept pre-made photos, they take a digital photo and it's attached to the application in the computer right away. Well, not right away, it took 3 attempts because the application they use it a bit buggy
This next and final part takes about 3 months. I'm not sure if there is any rush processing available, I was not in a rush. I'm actually going to pick up my passport next week.
Good luck!