Chantel

Yes.  Or if the child is a minor the parent can apply for the U.S. passport immediately after entry.     This can be completed at most post offices

Question.  Is your friend a spouse of a USC or is this a different type of immigrant?   Spouse of USC would only require the USC spouse to file the I-130.
 
For other types of immigrant I would lean toward starting the I-130 petition as a green card holder, then if the processing is still ongoing when your friend becomes a USC contact USCIS to change the status and visa type, at this point things will speed up  .     
 
If the parent is a USC when the child enters the USA the child will obtain citizenship automatically at this time.

Theoretically,  yes.