Perche

Naples is a beautiful city. Try to stay at a hotel on the Bay of Naples, on Via Partenope. Walking up and down Via Partenope and stopping at one of the open air bars for a prosecco at sunset will definitely put you into the dolce vita mindset of Italy.

There is debate about which place makes the best pizza in Naples. However, I don't think there is much debate that the best pizza in the world is made in Naples. In my opinion, Sorbillo is the best. It is in the Old Town, on Via dei Tribunale. When you see Via dei Tribunale, it will look like what you see when you rent an old Sophia Loren movie. It is not like Rome, Florence, Venice, etc. Old town Naples will probably fit your mental image of what Italy is supposed to look like, more than any other city that you might visit.

From Via Partenope to Via dei Tribunale is walkable, but you'll get lost, so I'd recommend a cab each way. There is actually a new Sorbillo on Via Partenope. It was opened by one of Gino Sorbillo's sons. I haven't eaten there, but I've heard good things. If it's half as good as his father's pizza, it will beat the pants off of anything you'd get in Milan, Florence, etc, but I'd stick with the original Sorbillo on Via Tribunale because the old part of town is worth seeing. http://www.sorbillo.it/it/.

Don't just eat any pizza. Naples is famous for the quality of its pizza, so there are a certain number of shops that have opened up and will sell questionable pizza for tourists. To defend their reputation, pizzaioli Napolitani started an association that guarantees an authentic pizza, Verace Pizza Napolitana. Look for the symbol. There are some places that are not part of the association that are better than places with it, but if you only have one day, you don't have time to experiment. http://www.pizzanapoletana.org/index_eng.php

Of major cities the only pizza that can even come close to pizza napolitana is found in only a few select places in Rome, and Roman pizza is completely different from pizza napolitana. It's a different thing altogether. You will have huge lines for pizza in Naples. The only problem is, Napolitani don't really do lines. You just force your way in. To avoid this, get there a little before they open.

Extending off of Via Partenope is Via Toledo. It takes you through amazing Piazza San Carlo, and down the main shopping street. It's definitely worth a walk.

In sum, if I had one day in Naples, I'd stay on Via Partenope, take a cab and grab lunch at Sorbillo on Via dei Tribunale, take a cab back Via Toledo (even though it's only about a 3/4 mile walk), walk down Via Toledo through Piazza San Carlo, admire the Piazza and the opera house, and be at the Bay as sundown is beginning. You will eat one of the best meals of your life, and you will see some of the most amazing sunset scenery you will ever see. After nightfall walk over the bridge from Via Partenope to Castel dell'Ovo. It will be obvious where it is when you are there. You can't miss it. Stop at an open air bar there, and enjoy the fresh air and scenery of the Bay.

Naples is old. Especially around the old town. It gives off a spooky vibe, but it's just a vibe. There's nothing to be afraid of in Old Town, on Via Toledo, or on Via Partenope.

Whatever you do, do not eat the pizza at Pizzaiolo del Presidente in old town.