From the information I gathered you have to go the Termini Station in Rome and then back out to the cruise port in Citivechia.
Maybe Hank or one of the other Europe "experts" can answer for sure. But when I asked about going from the port to the airport I was told that we would have to go to Rome and then back out to the airport. The train from Rome to the Port is supposed to be in the $15 to $20 range per person.
I decided on just paying for the cruise line transfer for the one direction. You might consider checking into a taxi or limo to get from the Hilton to the Port. Might be more expensive but less hassle.
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