I had an amazing time in Naples at Melania's apartment. She is a great host, very kind and very welcoming, she will tell everything about the city (she also provides guides and maps).
The apartment itself is just enough for a couple, even a third person. The location is ideal as it's in the heart of the old town, so you can do everything by walk and experience the real Napolitan life! Loads of shops around for food shopping, loads of restaurants, and quite a lot of museums too. Also the Duomo is barely 10 min walk and it's absolutely stunning! The main train station Garibaldi is 10 min walk as well and it will take you to Pompeii, Ercolano, Sorrento, the Vesuvius, Roma, etc.
Naples is a great city, so many things to do, transports and food are cheap.
I was traveling on my own and I never felt unsafe at all, so just go and stay at Melania's! Have fun!