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You should definitely read the "How to's of Train travel in Italy" (the first sticky post in this section of Cruise Chat. It will give you a bunch of information and after you read it you will be able to use the trains and subways like you have lived there.
To answer your question, the metro is Rome's subway system. The train from the airport is not integrated into the subway but you can take the airport train to the Rome Termini station and transfer to the Metro right there in the termini station.
The trains and Metro do operate on Sundays. however there are sometimes differences in the schedules on weekends and holidays. Also they are subject to worker strikes which can seem to happen without notice at times.
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, Magic 1/14, Freedom x 2 6/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013