See our Guide to Train travel in Italy in our FAQ section. Lots of great information in there.
Yes, there is an overhead compartment much like an airplane. There is also a larger area for suitcases near the train car entries.
Not sure what your definition of the "best way" might be. But the train beats the cruise ship bus (especially if you are staying in Rome before the trip). From the airport you have to travel into Rome and then out to Civ but its probably still faster (if you get the express trains) than taking the bus.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007