Not that I'm pushing the idea but you can do Rome> Pompeii Scavi in far less than 3 hours one-way, if so inclined. As mentioned above, you take a "fast" Trenitalia or Italo train to Napoli Centrale (abt 70 minutes) and then the Circumvesuviana commuter train from Napoli Garibaldi to Pompei S. Villa Misteri station (about 35 minutes). The scavi station is just outside the main entrance to the excavation (Porta Marina entrance).
Napoli Centrale and Napoli Garibaldi are connected; you walk inside from one to the other. As with all 'fast' trains, you'll generally be able to land some savings if booking in advance but the least expensive tickets are non-changeable and non-refundable. Circumvesuviana tickets cannot be purchased in advance but they're really cheap; just purchase at the station.
The Circ. is an unlovely thing (think 'L' trains in Chicago) but it's what the locals use and it gets the job done. Just keep a tight control of your valuables...which shouldn't be difficult as you wouldn't be traveling with a lot of baggage, and which you should do sightseeing pretty much everywhere anyway.
You want the Naples> Sorrento Circ. line; schedules here (scroll down for trains heading back to Naples):
https://www.eavsrl.it/web/sites/default/files/allegati/Orario%20Napoli%20Sorrento%20con%20Pozzano%20e%20Scraio%20e%20viceversa.pdf
And an abbreviated 2019 version (middle stops tourists generally don't need left off chart) to get a handle in English:
https://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule
And some info from Man in Seat 61 on how to do all of the above, although he doesn't note that you can also use Italo for the Rome> Naples leg, and you can buy tickets directly through Trenitalia or Italo sites versus from italiarail.
https://www.seat61.com/places-of-interest/day-trip-to-pompeii.htm