One day to see Pompeii or the Archeological museum

No contest weather permitting go to the archeological site of Pompeii or do both

Any train option operated by Italotrain or trenitalia's Frecciarosa that takes 1 hour direct to Rome is the fastest train. There are 3 or 4 an hour. For sure the earlier you buy the lower the price.

My. Choice would be the arch. Museum and I have even to both.
The museum is extensive. Read up on each and then decide.

Maybe consider backtracking (only about an hour) to Sorrento and seeing both sites in one day while leaving your bags at your Sorrento lodgings. We saw the Naples museum on the way to Rome but left bags at Napoli Centrale and then booked a train to Rome once we got back to the station and colected the bags ... going to Pomeii without bags was much easier. You could take an early train the next morning to Naples and then go straight to a Rome train, probaby get to Rome by noonish.

I recommend Pompeii.

If you can do both, that would be ideal but if not, Pompeii is the not-in-a-vacuum experience, if that makes sense? While both scavi and museum are valuable, seeing firsthand how the city was laid out, architecture/decoration that's still in original place after several thousand years, and Vesuvius rising ominously in the distance puts things into context, IMHO.

Seeing Pompeii is a wonderfully unique experience and well worth it. We went to both Pompeii and the Archeological Museum in Naples and saw the museum on the day after we saw Pompeii. The museum has many artifacts and murals from Pompeii, but if we had not actually seen the Pompeii site I do not think they would have been as meaningful. We stayed in Naples one night before a flight in a hotel on Via Partenope a pedestrian area by the sea.

Everyone is going to have a different experience. For me, the museum was FAR superior to Pompeii (heresy to some, I know, but you asked).

For trains, look at Trenitalia https://www.trenitalia.com/en.html and Italo https://www.italotreno.it/en (the two companies that operate fast trains on this route). Even in the English versions of the website, you have to use the Italian city names (Napoli and Roma).

I take it that you’re staying in Sorrento, but are considering heading to Naples for one night, then taking a train to Rome on the second day. But on the first day you’re wanting to pick between either the outstanding museum in Naples, or the physical site of ancient Pompeii. If only one is possible, the museum would give you the most significant experience. Once you reach Rome at the end, are you spending the night, or departing Rome that last day?

What we did 6 years ago, with more than just one day: we stayed in Piano di Sorrento, next door to Sorrento. One day we visited Naples for the museum and a pizza pilgrimage. The museum was wonderful, but the neighborhood in Naples we walked around was filthy, with trash and broken glass littering the sidewalks and streets. The city itself was an outdoor graffiti museum, as most every outdoor wall had been vandalized with spray paint. We got our pizza and then got back to the safer Sorrento area. Naples is a big, big city and some areas might be more comfortable, and maybe a lot pricier, but a little of downtown Naples for a day was enough, and not a place to stay for the night.

We did visit the Pompeii site on another day, and it was great being where the city had once been alive with citizens of Ancient Rome, walking inside the surviving bathhouse, seeing the villa with a reproduction of a mosaic floor, partially covered with leaves and dirt (the original, clean mosaic is in the museum in Naples), and catching a view of Mount Vesuvius from a street that had gotten buried in its eruption in 79 AD. But if the choice was between the excavated site with bricks and marble but few artifacts, and the Naples museum full of art and genuine artifacts, the museum has the goods.

There’s also the nearby smaller but fantastic archaeological site of Herculaneum (Ercolano), buried in the same eruption, maybe in better surviving shape than Pompeii. Maybe go to Ercolano instead of Pompeii, maybe the museum in Naples plus Ercolano the same day? You’d have to figure out your timing and transportation if you fit in both.

Then to get from Sorrento to Rome at the end, we took the Circumvesuviana train to Naples in the morning, then bought tickets at the Naples station for the next train for Rome. Arriving at the west side of Rome in mid-afternoon, we were a short walk from the archaeological site of Ostia Antica, not buried by a volcano, but buried in floods and now a great recovered site. See real, original mosaic floors there, a huge former public toilet complex (impressive engineering), and a tavern that seems like it could just about open again without much renovating. If you saw the museum in Naples on the first day, then slept in Sorrento (or Naples, if you brought your luggage) the first night, you could get to Ostia Antica and Rome on the second day, thus seeing the both the Naples museum plus an actual site, but the site at the west edge of Rome rather than the Pompeii site southeast of Naples.

Pompeii. Museum’s are great but there’s no substitute for being in the actual place.

Pompeii. However, Naples is not dangerous for tourists, except maybe from the wild auto drivers. Yes, it looks run down in places, but we walked all over with no fear. Many American cities have far scarier sections. And there is no opiate crisis in Naples.

Edit:
https://community.ricksteves.com/travel-forum/trip-reports/surviving-naples

By the way, in 2016, there were:
7.9 gun deaths per 100,000 in California,
4.4 gun deaths per 100,000 in New York State,
14.3 gun deaths per 100,000 in Colorado.

Sometimes we don't worry about the right things to worry about.

https://en.wikipedia.org/wiki/Firearm_death_rates_in_the_United_States_by_state

This is a very good thread, but regarding Naples, (and correct me if I'm wrong) when Americans see graffiti, they think crime and poverty, but in Naples it is considered to be more like public art. Is that right?

I did both. Hands down Pompeii although I totally agree that as long as you avoid the motorists, Naples is perfectly fine.

If you have to choose, Pompeii. It's an amazing site. And I agree with all above that you should be able to find safe places to stay in Naples. Look at the RS guidebook or reviews on TripAdvisor, and be careful of the traffic. We stayed a few blocks from the archaeological museum (can't recall the place) and it was fine.

Have you considered doing the museum and Herculaneum, instead of Pompeii? Herculaneum is like a bite-sized Pompeii and can give you the flavor of the experience without overwhelming you.

Walking in Naples, we were nearly hit twice by glass bottles thrown by kids against the wall we were walking near, and shards of ricocheting glass flew by our legs. Gun deaths in the USA aside, and car crashes and rattlesnake bites in Colorado are other things to worry about as well, but Naples is not the safest, most comfortable place around - based on personal experience, not optimistic fantasy.

And if some Neapolitan’s Alfa Romeo gets graffitied, they probably won’t appreciate the free “art,” whether the offender is poor or affluent.

As I said earlier and Michael has just posted, as well, maybe the Naples museum AND Ercolano (or Ostia Antica?) might get you the most for your time, if you can’t get to Pompeii as well.