VAlexCross, there is no "bullet train" from Rome to Pompei. There is no direct train at all. You can take a fast train from Rome to Naples but there you'd have to change to a local Circumvesuviana commuter train.
It takes slightly over an hour from Roma Termini to Napoli Centrale, where you walk over the adjoining Garibaldi station to catch the Circumvesuviana. You will need to purchase tickets for that train when you get to Naples (they are very inexpensive) and they must be validated before boarding. Same for the return trip to Naples.
If you are traveling during the warmer months, the Campania Express - which is a little more comfortable train - also makes runs from Garibaldi to Pompei, Ercolano, Sorrento, etc. and has fewer stops along the way. The only issue with that one is that it has far fewer runs per day - only 4 or so - so to make the best use of time, the Circumvesuviana is probably your best bet.
You'd want to exit the Circumvesuviana or Campania Express at Pompei Scavi Villa dei Misteri station, and it's about 30 minutes or so from Naples. The station is right in front of the main entrance to the excavation. This is the way I'd do it so that you could spend as much time as you wish to exploring the site: it's HUGE and can easily take a big chunk of a day without seeing it all.
You could also have dinner in Naples before taking a fast train back to Rome. :O)