Pompeii and Sorrento

That would be an awfully fleeting visit to both sites, and you wouldn't have much time at either. Figure an hour to taxi from the port and then train to Pompeii. From Pompeii to Sorrento it's another 30 min by train. From Sorrento to Naples it's about 40 minutes by ferry. Of course none of this includes waiting time for the train or ferry to arrive. So best case scenario, you'd have about 3 hours to see both of these places. We spent an entire day in Pompeii. Sorrento deserves the same. I'd recommend that you just concentrate your time at one of these.

Seconding Cjean here: you just don't have time for both. Pompeii is closer to Naples, and it's a HUGE site so you'll need all of your time for just that one. You won't be able to cover it all in just a few hours but you could do enough.

You might also consider Herculaneum as a substitute? It's smaller, even closer to Naples, and a fair amount of visitors have preferred it to Pompeii as it's even better preserved. You still wouldn't have time to do both this AND Sorrento, though.

http://www.pompeiisites.org/index.jsp?idProgetto=2

https://www.tripadvisor.com/Attraction_Review-g670330-d195507-Reviews-Ruins_of_Herculaneum-Ercolano_Province_of_Naples_Campania.html

Sorrento is a nice town to use as a home base during a longer stay but the archeological sites are the reason for the visit, so I also would focus on either Pompeii or Herculaneum. If you have any additional time, then try to get to the Archeology Museum in Naples.

You can do it on your own with no advance booking necessary, or perhaps the cruise offers an excursion that guarantees to get you back to the ship on time. Also, plan for it to be a hot afternoon, unless the same-day weather forecast says otherwise. Admission tickets and licensed guides are both easily available at the site entrances.

That afternoon arrival seems a little out of sync. Rick-recommended company Mondo Tour does not have any shared tours departing in the afternoon. Their rates for a private service are prohibitively expensive for one person (e.g., I personally would not pay €260 for 4 hours with a car and driver plus €130 for tour guide in Pompeii).

As previously noted, a cruise-sponsored tour, if they have one, would ensure that you got back to the ship before it sailed (or the ship would wait.)

(e.g., I personally would not pay €260 for 4 hours with a car and
driver plus €130 for tour guide in Pompeii).

Me either, Laura! It's much less expensive to do on your own by train if you do the reading-up in advance. You do take on the responsibility of getting back to the ship on time, though.

Definitely a case where it is worth taking a ship's excursion, especially as the train to Pompeii does not always run to the timetable. Don't forget that it takes time after arrival for passengers to get off the ship, and you need to be back on board half to one hour before the departure time of 7.00 pm so you will be lucky to have 4-5 hours ashore.

With 1PM arrival, you should not do either Pompeii or Sorrento on your own. Sorry, but pay for the ship tour.

If you're on a ship with another 2500-3000 passengers, expect that you won't be on shore until 45-75 minutes after docking (it takes time for the ship to set up for debarcation and passengers will be lined up well in advance. Favored passengers (frequent cruisers, captain's club - that kind of thing) and ship's tours get priority. Also you'll have to be back on board about 45-60 minutes before the ship sails. With such a short time in port, I'd go with either seeing Pompeii with a ship's tour (the ship won't sail without them) or "settling" for spending the time in Naples. Visit the wonderful National Archaeology Museum there, where most of the artwork (mosaics, frescoes, sculptures) from the towns buried by Vesuvius are on display (many of the ones in Herculaneum and Pompeii are replicas) as well as other treasures. Then take the RS self-guided walking tour of Naples, which starts across the street from the museum.