Jackie, your hotel should be able to arrange a taxi or shuttle service for you, as suggested by Larry. If it were me, I'd bite the bullet, stay in the city center, and get up early. You can try to doze on the plane and you'll be trashed when you get home regardless, so half an hour really won't make that much difference. Not only is it cheaper to stay in the city, but you can enjoy your last night in Rome far more.
Sally, if my flight were one of the first ones out at 7am, I might gamble and go for the 2 hour window. When I flew out at 10:30am last year, I got to the airport at 7:30am and had plenty of time to kill & no lines. Since you will be leaving more towards the middle of the day, I'd allow at least 2.5 hours, if not the 3 recommended by Larry. Especially since you are driving back in and dropping off a rental car, I'd allow extra time for getting lost, delays in returning the car, etc. As mentioned to Jackie above, a bit more sleep that morning really probably won't have that much impact--and you can nap at the airport if you have extra time.