The best time to visit Naples is between March and May. Daytime temperatures hover in the 80s — perfect weather for the beach or the links. More importantly, if you visit during this sweet spot, you'll sidestep the winter price hikes.
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You'll find prices here far cheaper than they are in Rome, let alone Florence and Venice. Southern Italy is far less wealthy than the north, with the result that Naples has a much lower cost of living, and that translates into cheap accommodation, cheaper food, and more affordable attractions for visitors.
Naples, with its ancient streets of Neapolis and Quartieri Spagnoli, is a walkable city; you can explore everything from the royal palace of Naples to Castel Nuovo on foot.
One of the best times to fly cheaply would be late May or early June - just before school holidays start. However, be sure to avoid travel in and around Memorial Day as you may find prices are slightly higher at this time.