Lets run a time line as you may not have as much time as you think ---- And assuming that you don't have several hundred euro to spend on a private driver for the day. That, of course, would be the best solution and easiest. And, assuming that you have too luggage that is common for cruise passengers. This is a range of times to consider --
7.30 - arrive. Most international flights are on time
9.00 - Depart on the Leonard Express to Rome Termini. It could easily take an hour to an hour half to get your checked luggage and process through immigration. The last time we were through immigration in Rome on a Saturday morning it was over two hours just for immigration.
10,00 - Arrive in Rome. It could be earlier. The train from the airport runs every 30 mins, taking 30 mins. But if you just miss it then it adds time. However, at the airport ticket office also buy a Regionale train ticket to Civit.... (so you don't have to buy it later). Regionale trains and the LE are open seating (no reservations) so those tickets ABSOLUTELY must be validated by sticking them in a time stamping box - a yellow/blue/green box near the station. Watch for locals doing it. If you don't the fine is substantial and payable on the spot.
On the left side, lower level as your face the front of the station is luggage check. Check your luggage at the luggage check. Sometimes the line can be long and slow.
11.00 -- Out of the station - walk across the plaza in front of the station beyond the taxi mess, the tickets for the various hop on/off bus are on the far, right side. Check the route and times that most appeal to you. With the head sets it a decent way to get a quick over view of Rome. The big disadvantage with be the tendency to sleep because of jet lag. Personally I would not recommend sitting down for a couple of hours in a swaying bus but for a quick overview of Rome you have no choice. I don't know the time lengths of the buses for a complete loop but assume it would be in the range of two hours.
2.00 pm Now we have a problem as we running out of time. If you have to on ship by 3 pm you probably don't have time for the bus. So maybe you do an hour or so walking loop. If you really need to be at you check in by 3pm you need to be back to Termini by 1pm to get your luggage and find your train's platform to Civit.... It is basically a commuter run to Civit... with three or four trains an hour so find the next R train to Civit.... There will be a couple of Intercity (IC) trains on that route but they require a seat reservation so your Regionale ticket is not good on an IC train. Find the next R train, validate the ticket and get on. It will take about an hour to get to Civit....
When you arrive at Civit.. - very small station -- you will see the ocean and probably your ship to the far right. The station is about a quarter mile or a little more to the port entrance. Most people walk it. Just follow the flow. There is a shuttle bus between the station and port entrance - probably 1.50E or more these days. Taxi are available but be prepared to pay a premium for a quarter mile ride. At the port entrance a shuttle will take you to your ship's check in. The area could be very crowded especially if more than one cruise line in port.
The pointing I am making is that your time in Rome will be very limited and personally would not make much of an attempt to see Rome unless I could get to Rome prior to about 9am. You could take a taxi to Termini that would be quick - 48E flat rate. And you should consider reducing your luggage.
We have made that run five times so we know it well. Our luggage is three (total) carry on size bags so we can move fairly quickly.
Good luck.