Schengen visa: It is 90 days in a 180 day period, so you can stretch your allotted 90 days out over a period of 6 months by flitting in and out of Schengen. You don't need to be outside of Schengen for 90 days before you can reenter, you can easily return after a few days. You would only need to stay outside of Schengen for 90 days if you wanted to restart the 180 day period, thus getting another 90 days visa-free. Before the Eastern European states became full Schengen members many people did some creative jumping around Europe that allowed them to travel for up to 1 year visa-free. This is still possible, but with Switzerland becoming a full member too people are pretty much limited to the UK, Ireland, North Africa and Turkey for this.
Pompeii (italian spelling "Pompei", under which you can find it on google maps) is definitely worth a visit. It is very close to Naples (30 min by metro) and conviniently located for trips down the Amalfi coast. (Better than Naples in fact.) If you intend to go in spring (March/April/May) it would be a real shame to miss the Amalfi coast. But you can of course just as well stay in Naples. Main reason to stay in Naples is bc Naples has more options for budget accomodation. In Pompeii the cheapest places to stay are the camping sites. There is also supposed to be an official YHA/HI youth hostel, but I cannot find any information online. It might be that it no longer exists. So without a tent it looks like you pretty much have to stay in a hostel in Naples or in Salerno.
Maybe would it be better to just skip that whole route altogether and fly straight from Lisbon to Italy?
Good luck finding cheap flights on that route. There is a reason I suggested to catch a flight to Germany or to fly from Barcelona to Italy. Portugal-Italy is not a major package holiday route, thus very few cheap flights exist. If you want to fly from Portugal you will have a better chance to catch a cheap flight to Southern Germany than to Italy. There are also more options from Spain and Morocco to Germany than to Italy. Maybe somebody from Portugal or Spain can help you with this though.