We hired a limo for 120 Euro (including) tip from FCO to Civitavecchia.
SS Norway (12 times) '85 - '97; Radiance OS 06/01; Island Princess 11/04, Grand Princess 11/04/05, Golden Princess 7/21/06, Golden Princess 2/24/07, Explorer of the Seas 12/23/07, Island Princess 11/15/08
The least expensive way to get to the port is the traain. The train from the airport goes to the main station and there you transfer to another train to Civitavecchia. Total cost shoul be about 15 euro's each., You need to have enough time. See www.trenitalia.com for schedule aand cost.
Carnival will also have a bus that will be competitive with the private shuttles for 2. For more that two not so.