Why not try Ostia, the former seaport for Rome. Its right by the airport and I heard a new museum was opening there.

Also, if you do some reading beforehand, flying into the airport is wonderful. I had gone via AMS, so we flew down from the Alps all the way over the landscape of Cicero and Catalina and Milo...

Try http://www.ostia-antica.org/ to find interesting things right near the airport.

On the other hand, it suddenly occurs to me you may not be as fascinated by early Roman history as modern Italy...in which case Ostia probably isn't of much interest (but still probably better than the airport!)

Actually, the Airport website http://www.adr.it/default.asp?L=3 has an article on the Roman Ship Museum right by the airport. Pass up a run thru the forum (its uneven ground anyway - always twist my ankle there) and stoll thru the museum and relax!