The "best" and the "most economical" are not the same in this case. The best would be to take the cruise transfer. A taxi/car would be very expensive.
The most economical way is to do it by train (train into Rome and then back out to the airport)
Assuming you got off the ship early you would have time to take the train but dragging your luggage on and off two trains and then through the airport.. the best way it isn't
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, Celebrity Silhouette 2013