One way to get from the port of Civitavecchia, where the cruise ships dock, into Rome or to the airport is by using private car transportation through a shuttle or transfer service. The following companies will transport you between Civitavecchia and Rome or Fiumicino or Ciampino Airport in a private car or van with driver and can be booked in advance.
Note: I have not had experience with any of these companies personally so cannot offer advice about them. If you prefer to pay in US dollars, you can book private transfer services online through an agency such as Select Italy (get quote or book direct).
Civitavecchia to Rome Private Transfer Services
More Rome Transportation Tips
For detailed information about how to get from the port to Rome, see Civitavecchia to Rome Transportation.
For transportation options to airports and other parts of Italy, see How to Get from Rome to...