Unless I missed something, your question needs more detail for anyone to reasonably answer it with advice that would be meaningful.
So, when is this all planned for ? Pompeii can get really hot, especially in the Summer months, so arriving at 2pm is probably not the best time in July or August, for example, but more reasonable in April or May. Also, depending on your interests, you can spend hours at Pompeii, so you would be getting a late start whenever you go.
If you are really talking about just one day, perhaps reverse the order. Go to the museum shortly after you arrive in Naples, and then spend the night and head to Pompeii the next day. Depending on how much time you want to devote to the visit, you might want to spend one more night in Naples, or just catch a later train to Rome.
Personally, if you are going all that way, why not consider heading towards Sorrento, Amalfi coast etc, rather than just going all that way, only to leave and not enjoy that part of Italy.
As for if the musuem is worth seeing, that would be up to you and your tastes, but if you are going all the way to Pompeii to see the ruins, then yes, the museum is worth seeing.
As to trains, just check the schedule. You don't have to change trains, just pick the right one. The train will likely stop in Rome, but will continue on. In the opposite dirction, from Naples to Rome the trip is about an hour
Everything you've asked is doable, but not sure many would do it the way you've proposed.
If you are still in the planning stages, consider other options.