Hi everyone. I've just received an expedite for an AP and since there's not a lot of information online about the process I thought I'd share my experience in hope it'll save someone some difficulty.
I submitted my AOS papers including AP application around 85 days ago in hope that I'd have enough time to get my AP before my academic and business appointments. After about 2.5 months I decided to expedite the process because I was getting concerned I wouldn't receive my AP in time for my flight. I should point out that simply submitting your AP documents (i.e. letter, evidence of emergency etc.) does not expedite your processing.
I called the 800 number to check on status and began the expedite process with a tier 1 representative. 4 days later I received an RFE with instructions to submit documents via fax. For the following 3 weeks I continually tried to submit documents via fax with very little success; out of the dozen + times I tried to fax the documents over a period of 3 weeks the office only picked up the fax once, and even then we didn't receive the confirmation receipt were were supposed to. I called the 800 number daily and talked to a number of tier 1 representatives. Almost all of them just told me to wait between 16 and 90 days, and if I received no confirmation of receipt letter I could call back again, which of course is far too late. Eventually I was put onto a tier 2 receptionist (after waiting 2 hours for a callback - 1-2 hours is the norm) that recommended that I post my documents to a PO box with the words "DO NOT OPEN IN MAILROOM" on the letter. We tried this, but even then 3 weeks after it was supposed to arrive we received no information as to wether or not it was actually received by the office processing my case. Talking to three other tier 2 receptionists (every staff member will likely tell you something different) it was clear there wasn't really anything they could do at all beyond submit a request for a status update, which itself takes at least 3 weeks to be posted out. In other words, from my experience the 800 number was completely useless other than to begin the expedite process, request forms, or check on status - most of which you can just do online. Not all of the staff were terrible, most of the T2 people were great and really understanding, but with two exceptions the T1 people were totally hopeless and frequently said things that were completely false.
Eventually I came across an article about someone's experiences going directly into a field office without an appointment and I decided to give it a try, in spite of the fact that all of the call center people I talked to about going in person said it wasn't possible and I should just wait (with the exception of one lady that told me I could try the field office in Jamaica, which is the wrong office). I went to the office near city hall. I first had to visit a kind of preliminary intake desk on the first floor where they checked my documents and asked questions, after that I was sent to the third floor where I waited for around an hour and copies of my documents were made and I was then interviewed fairly briefly. The clerk asked a few simple questions about my travel plans and mainly just checked over my paperwork. She did ask if I had my biometrics appointment, which I hadn't, so she scheduled an immediate visit to the passport office on the corner of Houston and Varick. I had read online that the biometrics weren't required for the AP and the call center staff also mentioned this also, but in my case at least it was required. I did the biometrics, which didn't take long, and went back to the USCIS field office, waited another while and walked out with my AP documents in hand. I think all together it took around 5 hours including travel to the biometrics.
It's worth mentioning that I was told the phone line initiated expedite process and the field office expedite processes are completely unconnected so it's possible to apply via both channels.
tl;dr: if you're applying for an expedite it's probably simplest to go directly to your local field office. If that isn't possible or you're rejected, then try apply by post. And after that, maybe contact the call center and ask to be connected directly with a T2 staff member.