We have
Rome in Limo booked for a post-cruise excursion this time around. It's private so it's a little expensive, but we've booked it with another family we met on a previous cruise that will also be on this one so it brought the cost down a bit. We haven't used Rome in Limo before but they have nothing but rave reviews. Others here on the DIS have used them, not sure if post- or pre-cruise, but definitely for excursions when docked in Civi for the day.
Now on the way to Rome, we have booked
Nicola Rome Transfer who is highly rated on Trip Advisor. I can't speak to personal experience (yet) but he's charging us 120 Euros plus 25 Euros an hour for the driver to stop (with our luggage secure with the driver in the car) at Necropolis Banditaccia, which I figure will be around an hour or so stop to look around at the tombs.
We considered booking with
My Day Trip. Someone recommended that transfer with stop in Ostia on Cruise Critic and said that their driver was great, but ultimately, My Day Trip assigns a driver (and it could be anyone) and I wanted to book direct and eliminate the middle man. (I'm middle-man way after a bad experience with Viator.) But that site is helpful to give you a couple of ideas of things that can be visited between Rome and Civi if you want to make a sightseeing stop.
In 2018 we disembarked in Civi and took the train. We sort of took our time getting off the ship in the morning and after we took the train and we ended up getting to Rome early afternoon. Our AirBnB host was available and ready for early check in - I had asked about it in advance. Not sure if you'd checked to see if this is an option or not, but some will work with you either for free or an extra early check in fee.