What a perfect time to be in Italy! We have been in Rome several times but never following a cruise. The best suggestion would probably be to ask the cruise line what transportation they recommend to get to Rome. Best to have your transportation completely figured out before arriving. You don't have a lot of time and it would be a shame to waste it searching for the train station or bus. I guess there are a few questions to ask: Is this your first visit to Italy or Rome? Will you have a lot of luggage? Are you a relaxed, slower moving traveler or are you more of the fast moving, fit-3-days-of-sightseeing-into-one type? Let me know a little more info and we can give you a few more tips.
I assume you are flying out of DaVinci/Fumicino airport. It is a good distance from the city. If you stay the night in Rome so you can really see some sights and enjoy the piazza night life (wonderful), you can take the train in the morning directly to the airport terminal. You will probably wish you had a few more days to really see Rome. Learning some basic Italian phrases will be very helpful and make it more fun for you.
Lise