We used Rome Cabs (
https://www.romecabs.com/). It was $150 each way, for four of us. That didn't seem bad given the long drive. They were really great to work with. We used them three times on the trip and they were always on time with easy to follow instructions. The drivers were friendly. The one that took us into Rome from the airport was practically a tour guide too - gave us a pretty good history of Rome and pointed out various sites as we made our way to the hotel. The drivers to and from the port were great too.
Have you been to Rome before? I have a few restaurants that we loved that I could post.
P.S. If you haven't been before, don't miss St. Peter's Basilica. That was the surprise tourist site for me that blew me away. I have zero expectations for it and it turned out to be one of my favorite sites. And so many little basilicas with masterpieces just on display for anyone to see for free. An unbelievable city. Man I loved that trip. I hope you have a blast!
Added: Many took the trains out to the port too. You do risk strikes or the like, and the walk from the train to the ship seemed long. But it is easy to do from what I understood from our group. For me, it was worth the extra money to have door to door service and no stress about getting to and from the port.