See When to go to Neapolitan Riviera for more general advice about weather variations and what's on.
If you’re planning on visiting Pompeii or Herculaneum it’s better to go in spring or autumn: there is little shelter from the sun on the archaeological sites and walking around under the full force of the sun can be exhausting. May is a busy month thanks to the ‘Monuments in May’ cultural festival when the palazzi, archaeological sites and churches are spruced up so that visitors can be shown around on open days. For information see www.comune.napoli.it.
June, July and August see temperatures soar into the 80s and, particularly in Naples, it can get pretty humid. In August many residents of Naples decamp to the islands where the sea breezes make temperatures more bearable: a spa break on the island of Ischia or spending lazy days on the beach in Capri, or perhaps just drifting around on a boat (remembering to put plenty of sunscreen on of course), can ease the summer heat.
'Estate a Napoli' is the name given to two months (August and September) of cinema, theatre, music (from jazz to classical) and spectacles that take place in and around Naples. For information see www.comune.napoli.it.
See When to go to Neapolitan Riviera for weather advice at this time of the year.
The Christmas period is a very special time in Naples. Putting up a Christmas crib is traditional and the street of the presepi (nativity scenes) – Via S. Gregorio Armeno – is especially busy as families go to buy a new figure for their Christmas 'crèche'. New Year (Capodanno) is usually accompanied by spectacular firework displays. For those who like their fireworks, it's heaven; at New Year, the entire city seems to explode (it has been described as looking and sounding like a war zone) as individuals 'arm themselves' with fireworks and 'bombs' and take to the streets. But as well as the 'ad hoc' fireworking there are always spectacular displays, such as 2009's at Castel dell'Ovo.