Pompeii

Does that mean you have to be onboard at 4:00? That gives you 3.5 hours max, not long enough to get to and from Pompeii or to explore a site as large as Pompeii. Even 4.5 hours is slim.
However, you could get a taxi and go straight to Herculaneum, a more compact, but very interesting site where the town was buried. https://ercolano.beniculturali.it/ You can hire a private guide at the entry (150 euros) and then get a taxi back after 2 hours.

Does the cruise offer Pompeii as an excursion?
If they do that’s your safest option
If they don’t that’s telling you there isn’t time

The archaeological museum in Naples is very good and has some of the best items found at Pompeii.

You may have to settle for the Archeological Museum in Naples.

However, you could get a taxi and go straight to Herculaneum, a more compact, but very interesting site where the town was buried. https://ercolano.beniculturali.it/ You can hire a private guide at the entry (150 euros) and then get a taxi back after 2 hours.

Herculaneum is a good alternative. It's also fairly closer to the port of Naples than Pompeii

The Circumvesuviana train (which does not leave from the Port of Naples) is just not reliable enough to do this excursion on your own. For a few hundred Euros, you might share a car service with another couple, but the risk is still on you to be back onboard, with plenty of traffic, construction, and Southern Italian infrastructure neglect dangers. What is the next stop of the cruise?

Naples is an historic place, with plenty of art, but it's a strangely short port day. Pompeii really is the attraction. Presumably the Executive Penthouse-Butler Suites get to leave the ship first.

Because Herculaneum is a 15-minute downhill walk from the Circumvesuviana, I don't rate it ahead of Pompeii for a risky "independent" (i.e. public transportation) excursion. Pompeii Scavi is right across from its Circumvesuviana station. You did not give the month of travel. This is a very, very hot and sunny location for much of the high season. And Pompeii is crowded. Most posters here recommend a minimum of four hours for Pompeii, and lots say they spent 8 hours there. We did.

Thank you for all of the tips. First , it turns out I was wrong about the return time. We have to be on the ship by 6.
The train sounds pretty sketchy so we're thinking about a taxi. Viator tours are all at the wrong time and the excursion from the ship is sold out. I will keep checking that to see if someone cancels.
The next port after Naples is Sicily.
Any advice or info about the taxi service?

We dock at 12:30 and leave at 5:00.

That seems an oddly short stop.

Are you sure the boat will be at a dock and not anchored off shore and use shuttle boats for passengers to reach shore? That would be an even greater intrusion on your useful time to tour.

As already mentioned, see what shore excursions the boat is offering to gauge what sort things you can expect to be able to do.

Here’s the taxi rates
https://www.napoliunplugged.com/wp-content/uploads/2018/03/Tariffe_taxi_2017.pdf

Viator is a reseller-doesn’t run any of tours they sell
Look for the actual tour company or hire a private driver
There are many mentioned here

Here is one we used for transfers in the area
https://www.topexcursionsorrento.com

Print out the taxi sheet to show the driver, in case he feels "entrepreneurial" and tries to charge you for passenger count or shoulder bags. It would also be wise to avoid haggling over splitting the fee in front of the driver. You want to look like one party. You need a couple of phone numbers in case there are no cabs when you come out of Pompeii, or if the driver that promises to come back at X:XX makes other plans.

Not trying to sound scary, just being realistic about travel annoyances! Certainly look for the train schedule, just in case. And find out where the newsstand (?) that sells the train tickets is. There's unlikely to be a human in the station selling tickets.

It's hard to find out in advance, but it's useful to know if the ship will be tied up to the shore or if it's a tender port, which takes longer.

Do yourself a favor and use the search box top center to read other topic posts here about Pompeii. About half of the posters here don't hire a tourguide, and love Pompeii anyway.

We used Askos for 3 tours in April 2022 - a year already. They also offer shore excursions.

https://www.askostours.com/destination/pompeii-mount-vesuvius/

It's very strange that your cruise ship doesn't arrive until noon. If it comes earlier, you might consider the CitySightseeing bus that leaves from the cruise port directly to Pompeii. It would be a lot cheaper than a taxi or car service. See https://www.city-sightseeing.it/shuttle-bus-naples-pompeii/