This museum is hard to define. While I called it a Cultural Museum, it is also a History Museum and an Art Museum for Ancient Art.
This is Naples' big world-class museum. Its collection of ancient art is unparalleled. Upon entry you can visit a great collection of Roman artwork, possibly the best in the world, save that of the Vatican.
At its core is the Farnese Collection, which contains some amazing art which was excavated from the Baths of Caracolla in Rome. Indeed some of this sculpture was excavated while Michelangelo was working on the Sistine Chapel in Rome, and he commented on it in letters home to Florence. If you spend some time looking at the Farnese Hercules, make sure, later that you look for similar figures on that ceiling.
Besides Hercules, the Farnese Bull is a wonderful statue. Room-sized, it is a depiction of a woman being chained to a bull as a punishment for disrespecting another woman. After you've been awestruck by the grandeur of the piece as a whole, make sure you check out the incredible detail of the sculpture, the animals and objects and the plants sculptured on all sides of this incredible work.
Also on this floor is the jewel collection which includes some incredible cameos including the Farnese bowl, an amazing large cameo bowl, which is ancient and delicate, carved with amazing skill on both front and back.
Upstairs, an extensive Art Collection from the excavations at Pompeii and Herculeum sits. There are wonderful mosaics, most of which look like paintings. One of the most notable of this is the huge battle scene with Alexander the Great, which came from the House of the faun in Pompeii. Remember this mosaic when you visit Pompeii.
If you want, and if you ask when you come in, you can get access to the "Secret Cabinet" in the museum. This is a room full of erotica, from rough phallic sculpture to a detailed sculpture of Pan with a she-goat, to frescoes of the "menu" on the walls of Pompeian bordellos, to vases and mosaics, even wind chimes, all manner of erotic art is here. I'll put up my photo of the painting of the three graces, which is the tamest of this. Much would be inappropriate for public consumption.
Everyday cookware and various implements, including jewelry from Pompeii and herculeum are displayed, and there are innumerable other exhibits.
Another highlight are couple large depictions of the Pompeii site; a diorama and a map on the wall.
The museum has a well-stocked bookstore, and a beautiful open garden in the center (closed when we were there) with much beautiful sculpture.
Getting there via the subway, the closest stop is Piazza Cavour. There are also a number of busses and the like which stop there.
Tip: I recommend doing this museum BEFORE you go to Pompeii. It provides and excellent introduction.