Yes, it part of the city of pompeii, although it was built just outside the city walls, but it is still with in the Pompeii site, and as such can be viewed with in your days entrance. it is well worth the extra walking to see the fabulous wall paintings

The door by the Villa dei misteri is only an exit. You have to enter through one of the entrances. The closest one is downhill on the left hand side of the road when exiting the Circumvesuviana station. You will then have to make your way to the villa. Go early in the morning before the tour busses arrive

Cruise ships generally have excursions to Pompei. Be prepared for a hot few hours with lots of other Visitors. I would not recommend going on your own there. If the cruise offers a trip to Paestum take that in preference to Pompei. The site is much quieter and the temples are out of this world. If you Stay in Naples There are lots of sights to see. I recommend a trip up to Capo di Monte palace. Easy to get to by bus or taxi. Go in the morning and be sure you see Andy Warhol's Vesuvius and Carravagio' flagellation which is breathTaking. From there go to see the Catacombs of San Gennaro, which is a short bus ride from Capo di Monte back to Naples centre. Check opening times on line for both attractions. If you don't have that much time stay in Naples and see the cathedral, the historic centre and the underground world of St Lorenz Maggiore, which takes you back through the Greek, Roman and Medieaval history of the site. Be very careful around the train station and on public transport in Naples. The pickpockets are masters of their art!! For a more energetic day You can get a 1 day ticket that covers the whole of Naples by bus and metro M 2 mto Pozzuoli, which has a fantastic Amphitheatre and Roman market as well as the Solfatara, a crater that still sends out steam and bubbling mud. Around the harbour there are lots of nice restaurants and if you are there before lunch time try and get to the Rione Terra, the old part of the town. The duomo is partly a roman temple an is a great place to visit but it closes at lunchtime so check opening times of the places you want to see.