I have two Naples recommendations for you.
Go to the palace of Caserta (by train). You will likely have the gardens and the palace largely to yourself. They are similar to Versailles, not quite so grand, but very interesting (more heavy Neoclassical/Empire than Versailles).
Try to also go by train to Pozzuoli. It is harder to arrange, but loads to see there. There is a vulcanic site, a lake created by an earthquake which is said to be the entrance of the underworld, the shrine of the Cuman sybil...
But Caserta is worth it, believe me.