We have made the trip from the port to both Rome airports.
The first occasion was in 2016; we had a private transfer to CIA arranged privately as part of our pre-cruise planning. The cost was €140. It was a professional service using a Merc E Class.
The second occasion was in 2019; we purchased the transfer from Guest Services on Infinity to FCO. We did not know the cost at the point of booking, which was $75 per person; we felt this was expensive. It operated like a shore excursion and was only suitable for flights departing after 13:30. Whilst the bus was not full so that one could spread out for more comfort, we would not retake this option because there is no Celebrity Rep involved. Therefore if anything had not gone to plan, we would have no way of rectifying the situation other than to resolve it privately. It was just a bus with a driver. It was a nightmare resolving an errorornous minibar charge after this cruise with Celebrity, I wouldn't want to be negotiating a resolution over a failed transfer where I am out of pocket after the fact.
In future occasions, at Civitavecchia, we will arrange a private transfer.