Travel from Port to Airport in Rome

Take the train to Rome and the train (Leonard Express) to the airport. However, fastest and cheapest do not go well together. The above route will be around 30E/person give or take a little depending on which train you catch to Rome. And take between two to three hours including waiting to catch trains. A taxi would be in the range of 100E but door to door and less than an hour. See RoneinRome.com for details of getting between the port and Rome.

The port is in a port city called Civitavecchia, about 40 min drive north of The airport Rome-Fiumicino (FCO). The fastest way is with an airport transfer service (shuttle van or limo). The cost is however high. At least 120 euro for up to 3 passengers. The cheapest way is by train. Take the train from Civitavecchia to Fiumicino Aeroporto. It will require a change, generally at Roma Trastevere station. The total train journey time vary from 1h 22min to about 1h 50min. There are very frequent train options, about 1 to 2 trains every hour. The cost is only 11 euro per person (generally).
Use the website below to find schedule. Enter from Civitavecchia to Fiumicino Aeroporto. For date enter a day this week (timetable is shown for the next week only), for time enter your time of arrival at dock. The CIV station is about 1/2 mile from the port, so take a taxi: Www.trenitalia.com

Adding on to Frank and Roberto, the only direct way to get from the Civitavecchia port to Rome FCO airport is private transportation. You will see lots and lots of taxis, limos and a mini-bus or two just waiting at the port for passengers to the airport. It's not cheap. Your more reasonable alternative is a 90-110 minute 2-train journey from the Civitavecchia train station to Rome-FCO via a train change either at Roma-Trastevere or Roma-Termini. The fastest and cheapest routing is via Roma-Trastevere. 11Euro.