I have expereince of www.romashuttle.com
Cannot recommend them highly enough.
Picked up my relations on time from the port, waiting with sign as requested and sent large van.
En route to airport uncle realised they had had a 'senior moment' and had forgotten their passports......Carlo the driver was fantastic - did a u-turn, flew back to the ship and then zoomed to the airport to ensure they caught their flight.
At the airport he still quoted the original fare of €110 - considering he had done the journey practically 4 times...
Alternatively you use the cruise transfers which are approx £57pp each way, or you DIY which is what we do. Catch the Train from the airport to Roma Termini for €11pp and then catch the train to Civitavecchia for €9pp. From Civitavecchia it is a simple straight walk to the port entrance where you will find the free shuttle bus to take you to the ship.
I too have used romashuttle on 3 occasions there & back. Always on time with good large vans as well as smaller vehicles if needed. No payment in advance, just a credit card to secure the booking, then payment in Euro's at the end of each leg. Regard.
Hope that you are arriving previous day as if you are late they will not wait for you.
we always go a day early.
Celebrity do not offer transfers from CIA -you could catch the TerraVision bus into Rome and then the train to the port. I beleive the bus is around €.