Depending on what type of cruise you have booked you should have a coach transfer from the port to the airport. If not just contact your booking agent as there are many bus company offering transfers from the port to rome. It takes about 1 hour and 20 minutes.
There is a regular, reliable and enjoyable train ride from the port into central Rome. It costs minimal euros and is much cheaper than a taxi or the cruise ship transfer costs. The bus is another option but it takes a long time to get there. Go by train, the station is within walking distance once you get off the ship.