You need to be at the airport by 5:00-5:15.
At that time in the morning there will be zero traffic and a taxi will have you there in approximately 30 minutes.
I’ve taken early flights after staying in the city, but, when I spent only the last night in Rome, I stayed also at the airport Hilton, which is about 10 min walk from the terminal via aerial walkway. The cost of the Airport Hilton is not exorbitantly high. I’ve never paid over $250/night at the airport Hilton. I’m sure a similar hotel downtown Rome would be more.
By staying at the Hilton you get to sleep maybe 20-25 minutes longer, at best.
Whether you want to switch accommodations for the last night just to sleep 20 extra minutes is up to you. I don’t think I would if I were spending multiple nights in Rome before flying out. 20 extra minutes of sleep would not be worth the hassle of an extra hotel change. Not for me at least.