First, you will probably not have a total of 5.5 hours of lame-duck time. Assuming that the plane is on time, you will have to get off the plane, go through passport control (make sure they stamp it), walk through the luggage area, get your bags if you checked them, walk through the customs nothing to declare doors and then you will be out of the air side of the airport. All that (especially passport control) will be affected by exactly when you arrive, the number of people who are there at the same time you are, and how the repairs at FCO are progressing. It could take as much as an hour.
Once on the public side, you will need to get your transportation into Rome. That's likely to take at least a 1/2 hour travel time, plus however long it takes you to get yourselves organized and into whatever mode you choose.
If your apartment owner cannot let you check in early (be sure to ask, mine always has) or has no way for you to leave your luggage there, one fairly easy option would be to leave it at Roma Termini, however you get into town. This is the link for information about that: http://www.romatermini.com/en/service/detail/left-luggage. Of course that is not a free service, but might be well worth the money if you have no other option.
Walking around is a very good thing after the long flight. Try to eat lunch around normal eating times (usually 12:00 - 15:00). This may not be a problem for you. Rick Steves' guides will have lots of suggestions for places to eat. You can find more in the Travel Forum questions and responses for Italy.
I among others have recommended La Gallina Bianca. It's about a 10 minute walk from Roma Termini. Click on Location on the opening page (website seems to be a work in progress), then click on Locate Us for the map. You'll see how close it is to Termini. Zoom in and take the little man to in front of it so you can see what it looks like. My husband was amazed that the outside tables are actually on the street.
The time will fly by faster than you expect and by the time you finish lunch, your apartment should be ready. You can go back to Roma Termini, get your luggage and take whatever transportation you choose to your apartment. Get settled in and go grocery shopping for things you may need that you might not expect. Trust me, that will be fun.
It will be hard to resist taking a nap, but do your best, or at least limit the time. Then hit the streets in the area of your apartment and explore. You're going to have a blast.