I am researching this same issue for our cruise starting Sept 18. According to blogs I have read on Travel Advisor, there are two connections to make from the airport to the port of Rome (Civitavecchia--I think I am getting the hang of saying it, at last!)
first is a metro from the airport to the Termini in Rome. Termini is the "terminal", specifically Train Station from which one can travel almost anywhere in Italy. Take a train from Termini to civitavecchia. I understand the cost of both is in the neighborhood of $45 (30 Euro). A one-way transfer through Carnival is $75 pp. I find the convenience compelling for the $30 pp price difference, considering we will be jet lagged and have to be vigilamt for pick pockets.
You can get transfers from a hotel to the ship; you would have to arrange to get into Rome on your owm. They have central pick up points in Rome.
The one piece of info I don't have is whether when you arrive at Civitavecchia on the train how far of a schlepp it is to where the ship you want is docked. we could be looking at another expense there?
I will be glad to let you know our experience. We are traveling with another couple who negotiated spain earlier this year strictly on metros and train. they want to adventure frm the airport to the shipl We on the other hand are opting for the tried and true if non exciting transfer to the ship. We are stayig in Rome (tbd where) after disembarking so proably will opt for the adventure route then!