What are you going to do if someone fails to appear at the meeting point. I personally would not accept the job of being responsible for this arrangement. I advise you to run as fast as you can in the opposite direction. With so little time (say, 8AM-4:30PM) on the ground beside the ship, and with a train ride of slightly over an hour, that leaves you with less than six hours in Rome, if things go well. Your group really should take the "overpriced, rip-off, proprietary cruise ship" trip. Big bonus: If they are late, the ship will wait.
May is not remotely as hot as June, but it will still be plenty crowded. You might have to stand on the train, but I can't predict that. You have the sad duty of telling the "Vatican" fans that they will wait in the security line at least one hour, and another line to get into St. Peters, after their Vatican Museums tour. This is a prescription for disaster.
(I have been to Rome many times, but only once to the Vatican.) We found the walk from the ship to the train in Civitavecchia to be long, cobblestoned, and not that easy to follow. (Using our host here, Rick's, book.) You should seek out Rick's official "Rome In A Day" trip, maybe in his "Cruse Ports" book in your biggest public library, or possibly on the internet. But it's not hard to find posters here who found the walk trivial. Because our 320 passenger ship used the proprietary port busses, we had to find the right coach, and sit in it (no A/C ...engine off) until the driver's departure time (to three different cruises) from the traffic circle to the piers. For security reasons, you cannot walk on the piers independently.
If you go forward with this, one of the first things is to read our host's Train Travel instructions, blue menu top left,
https://www.ricksteves.com/travel-tips/transportation/trains
You'll go to the Trenitalia site and type in "Civitavecchia All Stations" to "Roma Termini". You'll see that (I think) there are at least two trains an hour, but they all make a different number of stops, and take slightly different times. That's important because you are syncronizing an uncertain boat disembarkation with a predicted guess at your train time, to meet an inflexible bus tour with many passengers who have nothing to do with your group. Do you use public transportation at home, ever? It's real change for some people to not be guiding their armored SUV from one mall to the next, under their own control!
I'm not suggesting that you must buy these, just a few Euros, tickets in advance, or at a discount. But in fact, taking the "first train", with 1743 of your closest personal friends who just got off the ship, is a challenge of its own. I'm talking about making a transcript of all the likely train departure and arrival times, and thinking about things. I'd hate to be stuck in a line of 30 people with iffy credit cards, at the train ticket machine! That's also an obvious problem with the local number-route busses, or even the provided cruise terminal shuttle bus.