(1) M.A.N.N. - i.e., the Museo Archeologico Nazionale di Napoli.
Arguably the single greatest collection of antiquities in the world, including the Farnese collection of monumental sculpture taken from the Baths of Caracalla and elsewhere in Rome, plujs the matchless frescoes and mosaics recovered from the various Vesuvian sites - Pompei, Ercolano, Oplontis, Boscoreale & Stabiae. Ridiculously cheap at 6.50 Euros.
(2) Take-out pizza.
Avoid anyplace that advertises itself to tourists. Strike out on your own and look for locals queing up. You'll be rewarded with the best pizza you've ever even imagined, starting at about 2.50 Euros for an individual Pizza Margherita that will probably be more than you can eat at one sitting, cooked fresh in a wood-fired oven while you watch.
New Years in Naples just has to be seen (and heard) to be believed. And even after you've seen (and heard) it, you still might not quite believe what you've just seen (and heard). The real madness begins at about dusk - 5 p.m. or so - and gradually escalates, for the next seven hours, until the big explosion at midnight - at which point you won't be able to hear yourself think, for about an hour or two.
No charge for this last. But be sure to pack some ear-plugs!