Can anyone recommend a shuttle service ... or alternative... to get to Rome's airport so early? Would it make sense to find a hotel (suggestions?) near the airport?
I did this same thing in May. Spend your last night in Rome! My B & B ordered a taxi a day or two ahead of time for us to take us to the airport in the early hours. It worked great. By having the hotel/b&b call for it, I didn't have to worry about the language barrier and/or misunderstanding anything.
The last COTRAL night bus leaves from the Roma Tiburtina train station at 03:45, stops at the Roma Termini train station, and then goes directly to Fiumicino. See here.