Well....
we did street food and sat outside at a pizzaria that was wonderful - I didn't write the name down though - it's in front of an art school and we were walking from the Archeological Museum back down to the dock where our cruise ship was anchored.
Definitely do the National Archeological Museum as a lot of the things you expect to see at Pompeii have been removed to this museum. There are also some wonderful statues so don't skip over them. It is located at the corner of via Foria & via Toledo. I just pulled our map and it's marked that we walked down Via Toledo. Then we cut over at either VS Giacomo or Via Imbriani - this will take you by some beautiful municipal buildings and a wonderful 'hall' with glass roofs and a nice gelato place in it - I think it was the Galleria Umberto I on Via Toledo. This walk also takes you through their 'Spanish Quarters' which is one of the more densely populated areas of town with laundry handing out on lines and graffiti on closed stores. If you want to shop - remember to do it early in the day or late afternoon because they close up the shops midday - all of them! But as you walk down towards the bay, you will come to the Royal Palace and the Castel Nuovo.
Last time we were there, we went back to Pompeii again - we always see something new and different that we didn't see the last time. Best guide is this wonderful guy from Sorrento that waits up by the ticket stands - I have pictures of him but can't recall his name - used him the first time and saw him again the last time. Nice looking gentleman with graying hair. He's great! Be sure to take in plenty of water, wear sunscreen and a hat. It gets hot in there.
Also, last time we were there, we didn't do it but I think they have added one of the open top double decker bus type tours through Naples which everyone said was nice. I will warn you - Naples in dangerous in every way - lots of pickpockets and gypsies although we have never had problems with them - just be aware of your surroundings and you should be okay. Traffic is THE WORST in the world. My husband covered his eyes as our taxi driver forced his way through traffic and mopeds and pedestrians just barely got missed. So watch where you walk and try not to cross streets any more than you have to. Plan your route.