Civitavecchia is about an hours drive from the airport depending on traffic.... As to costs google transfer to Rome airport and some of the best private transfer companies will come up..
However if you are looking to head for a Transatlantic flight all I can say is - FORGET IT!! that is cutting it far too close remember you have lots of queues to deal with and need to be there at the very least 2 hours prior to departure...Airlines even say 3 hours to make sure you clear all the new security.....
You would have to be off the ship by 8:00 am and be at the airport by 09:00am that would mean no hitches or hiccups at all!!!
You could stress yourself out and try but be ready to miss your flight and need to rebook for another day (the next may not even be available.) On the other hand you could enjoy another night in Rome, enjoy the sites and get to the airport the next day in peace and calm... The next question is which is cheaper splurge on a another night in Rome or miss the flight, pay penalties and need to pay hotels for another 1 or 2 nights???