The AliBus is your best bet on getting out of the airport and connecting with a train in Naples. 5 euros per ticket. It’ll take you to the port, near to trains. On the way back, skip the bus. Grab two people from the AliBus stop at the main train station’s (Garibaldi) main entrance, and head for a taxi some 20 meters away. It’ll cost you each exactly what the bus would, and it will get through the traffic a lot quicker than a large bus. Our ride at rush hour was about 20 minutes.
Pompeii: They have a free app – Discover Pompeii. With a GPS map where it’ll show you where you are, and items you can click to get an audio guide… it appears to be the same system you can purchase at the door for 8 Euro. Skip the guide, get the app. Entrance is 15 euros for the day, 18 for a three day pass. You will not see everything in one day… so maybe try going there earlier in the trip, and then you have tickets for a ‘filler’ visit if some other plan doesn’t work out. Last entrance is at 18:00 and 19:30 they close.
Vesuvius: Take the train to Ercolano. Turn left into the dead end street when you get out of the station. There you’ll find Vesuvio express that’ll take you almost to the top. 10 euro ticket to get into the park, 10 euros for the ride there and back. You’ll have two hours to get to the top and back once the bus drops you off. Don’t stop much on the way up, go to the end and then use the remaining time to stop where you want. Most tourists don’t make it all the way to the end, so you get a peaceful experience toward the end. The volcano was spewing a bit of smoke, and if you look closer on the ride up, you’ll see burn spots everywhere. Wicked.
Sorrento: Sorrento Mill <3 Got there at dark, so only got a birds eye view. There might be an option to get in, as the place is in what is considered a park. This is all the info I have, and i would love to learn a bit more if you discover it closer.