The hotels in Ostia are quite reasonably priced...but you'd need a taxi for such an early departure. Waking up at 3am? What time are you going to go to bed? Attempting to sleep in that sort of situation is all but pointless.
My advice would be to find a bar near Termini that stays open till 2 or so...then hop onto the night bus...Spend a couple of hours at FCO, check in and fly out.
FWIW, I was stuck at FCO on Christmas eve. Alitalia were kind enough to accommodate me at the Hilton so I didn't quite spend the night on a bench (though the Hilton was well dodgy), but I did see a decent number of people at the airport who obviously intended to spend the night there, and nobody in a rush to remove them.