Not from Napoli, but I may as well share my culinary pilgrimage that I just have to do every time I go there (3-4 times a year):
1)
Attanasio: just outside the main train station, you just have to go there to get a sfogliatella riccia (remember: "riccia", not "frolla"! they're both good, but riccia is perfect if you eat it right away, while frolla can still be good if you have to bring it home and eat it some hours later.
2)
Sorbillo: arguably one of the best pizza in the world.
Yes, now a true neapolitan will come here saying how some other pizza is better. Still. Don't care. This is awesome anyway.
Just be aware of the long queues at all times (and they don't accept reservations).
3)
Di Matteo: quite close to Sorbillo there's this other excellent pizza place; I don't go there for pizza but for "frittata di pasta", which is basically pasta leftovers "glued" with eggs and fried. Totally amazing.
4)
Friggitoria Vomero: an old but gold fried stuff place. Just try whatever, it's all great. (I prefer Di Matteo's frittata di pasta, but I guess it's just a matter of personal taste).
A couple of advices:
- consider that on Sundays or for most of August, Napoli is pretty much shut down for holidays. You'll still be able to find some great stuff, but likely not the best (ie. Attanasio is pretty much always open; Sorbillo and Di Matteo aren't).
- remember to go there after a couple of weeks of diet, or else it's gonna be REALLY bad.
