Having been on two cruises in/out of Civit.. and one that passed through, my first response is that you cannot do it. Second response is, you might but I would not try. And this is only if you are doing carryon luggage only.
Is the flight directly to the US OR to another European city for a connection to the US ? -- Very Important. If a connecting flight you might get by arriving as late as 9:30 although at that time of day a 9 AM arrive would be much better. If direct to the US you would need to be there by 8. Time from Civit.. to FCO via limo/taxi will be 45 min to an hour. That will be you only choice and it will be expensive. By train you would have to go into Rome and then to the airport. A good two, 2.5 hours total trip time.
When a ship docks at 6AM it does not mean you walk off at 6:02 or even 6:30. Ships have to be cleared by port authorities, etc. so it can sometimes be as much as an hour after the ship docks before passengers can depart. Unless you are doing carryon only luggage, the ships has to have time to get your luggage off. Next the ship gives first exit priority to passengers who have booked airport transfers through them. For everyone else the ship established a priority schedule for departure that is coordinated with luggage. If everything clicks you might get off by 7. And grab the first cab.
However, if you have only carryon luggage you can walk off with the airport transfers, grab a cab. That will be about the only you can do it.