I agree with the suggestion of the airport Hilton, although depending on what terminal your flight arrives at, you might need to take a shuttle to get to the hotel rather than walk. It's a good distance from the airport into downtown Rome, but if the hotel shuttle is free (which I can't confirm as we didn't use it), then take advantage of that and enjoy the history of Rome.
I can recommend
Life Restaurant in Rome; it is absolutely fabulous. It is near the Spanish Steps, not anywhere near the airport, but you'll be there for a few days, so you have time!
