The best kept secret about Naples is how much fun it can be. It is an exercise in excess, in every respect. Rick Steves notes that Naples is very Italian, very flamboyantly so.
Stanley Tucci loved Naples in his latest CNN miniseries, and you can see how exuberant the city and its citizens are.
We stayed away from the "famous" pizza places, preferring to do our own research, and were rewarded by incredible food at 50 Kalo, as well as other places all across the city.
By all means, consider taking a funicular from Centro Storico up to the Vomero and then traveling by bus along Via Manzoni to view the city from high atop Posillipo, and maybe take a train out to Pozzouli to see the 3rd largest amphitheater ever built. It was begun by Nero, and is wide open with no crowds.
Sadly, your visit will be too short to do many things, but the energy is contagious. Be aware, however, that if your plans change and you will be in town on December 31, the New Years celebration in Naples is unlike anyplace else in the world. Not for beginners, but definitely an experience. Safety wasn't a consideration on my last visit. It's my favorite Italian city.