Leaving Barajas Airport During a Long Layover

In Madrid you are still in the Schengen area so no control - you just leave the airport.

On your return to the airport you will go though security and there's a passport check as you're leaving Schengen. Customs are for things and you will not encounter this until you enter the US.

Depending on what airline you are using, you might have to go through two passport checks when you get back to the airport. First you go through security. Then you go to immigration (passport check). Then, at least on Delta, you go through a final passport/boarding pass check before going to your gate.

Take the express bus from the airport to Atocha Station, leave your bag in a locker there, and walk to Picasso's Guernica and/or Velasquez's Las Meninas depending on whether you're in the mood for newer or older art.

http://www.esmadrid.com/en/airport-express

Nice parks to stroll in either way.

This article explains the options to/from the airport. There are lockers at the Atocha train station where you can store your stuff.

Make sure you have both boarding passes when you check in at Naples. If you don't, you'll have to spend time checking in for the second flight. When you return to Barajas, you'll go through security which could be slow and passport control, usually short.

There is also luggage storage at the airport. I used it about 5 years ago when we had an 8 hour layover.

Before you store at the airport, check your terminals. If you aren't departing from the same terminal you arrive at, it may not be as convenient as the train station.