You can ask for a place to crush at the occupied Ex-OPG Je so pazzo, a huge recently squatted place in Materdei, ora make some friends in the young life meeting points (piazza san domenico, piazzetta nilo,) and ask if someone can host you for a night or if they know a good place for you to stay for a night.
The police wont bother you too much if you just sleep on the street (of course not in a really major tourist spot) but you should be aware of the people, as they may try to steal your stuff, even using violence sometimes.
The cheapest area of the town is the old city center, also for supermarket (MD discount and immigrants shops).
You have to try the real pizza from Naples but you don't want to pay more than 3 euros for a Margherita. Usually the prices you see from the outside menu are including the service but if you ask to take the pizza away it will be cheaper, around 50 cents or one euro less. Ask for it inside.
In Napoli busking is permitted by law everywhere without needing to ask for permits and stuff. The local people are not really likely to give you much money, you'll have better luck with tourists that hang around the city especially in the summer period.
Best locations are in the City center: via Toledo, Spaccanapoli, Via Chiaia and the lungomare (the walking road near the sea).
There are many buskers in the city, so expect some competition.