I love walking around Naples and every time I discover new places.
There are many associations who are improving social situations of some nieghborhoods in Napoli and organize guided tours.
One the best tours I’ve done was with the cooperative La Paranza into the Rione Sanità where I discovered the Catacombs of San Gennaro and San Gaudioso, amazing places dated back the 2nd century AD where you are able to visit the underground of Naples without suffering claustrophobia and see incredible frescoes of the 6th century. The Catacombs are accessible to all, for extra infos:
http://www.catacombedinapoli.it/en/ (guide included in the price of the entrance)
My suggestion is to visit the catacombs during the first tour of the morning so you have enough time to see other places or eat something.
If you don’t mind walking, from the Archeological Museum you can reach the Curch of San Giovanni a Carbonara: a church built into the 14th century, which hide the incredible funerary monuments of the family Angiò last members.
If you are not tired of monuments, I suggest to see The Treasure of San Gennaro: you will love this amazing collection of jewels which can be compared with the Treasure of the Queen of England. Don’t forget to visit the nearby Cathedral.
Another option for another day could be a tour above one section of the Greek- Roman City of Neapolis. You can start this tour from Bellini Square, where you can see the Greek walls of the Ancient city, and keep going to Via Tribunali where there’s a great choice of attractions.
I sometimes love walking into this street without visiting any place, just to enjoy the real lifestyle of Naples, with guys on motorbikes without helmets, the eggs seller on the street and the fast crowd that accompanies the sunny days with their loud voices. I stop for a babà and a coffee and walk down to the coloured street of San Gregorio Armeno, full of craftsmen who create nativity scenes and figurines (some of them with faces of famous people such as John Paul II, famous soccer players, Barack Obama, etc…).
In case you are interested in a place to visit in Via Tribunali, I higly recommend to see:
1. San Severo Chapel with the amazing statue of the veiled Christ: http://www.museosansevero.it/en/
2. The Church of Santa Maria delle Anime del Purgatorio, where neapolitan people pray for adopted skulls in order to give to their souls the ascension from Purgatory to Paradise: http://www.purgatorioadarco.it/
2. Underground Naples, a visit to the ancient Greek aqueducts, not recommended for people who suffer claustrophobia: http://www.napolisotterranea.org/en/naples-underground/ (guide included in the price of the entrance)
If you wish to help local charities but also to admire one of the most beautiful religious complex, I suggest you to visit the Church of San Gregorio Armeno and its cloister, where the nuns help poor children and give to some of them the possibility to study: https://it.wikipedia.org/wiki/Chiesa_di_San_Gregorio_Armeno
Those are just some suggestions about what to do in Naples and I will be happy to give you some extra infos,so do not hesitate to ask!