well, if he says it's close to Narita my guess is that he's at Atsugi (which is not in Atsugi actually, but between Ayase, Ebina and Yamato towns) or Yokosuka, which is actually in Yokosuka. Neither Atsugi nor Yokosuka bases are 'near' Narita but they are closer than the other choices and it only takes 1 1/2 to two hours to drive from either of those bases to Narita. Relatively close, Japan-wise.
So first, you are within train distance of Tokyo, so you can do day trips or even go up after work. Something I'd recommend would be this, the New Sanno Hotel, which is for military people:
http://www.thenewsanno.com/ It's really nice and very reasonable. It's hard to get reservations on weekends but you can try to see if there are cancellations or you can try for a room during the week. Summer months are also hard because kids are out of school. Still, if you can get there, it's a great place to spend a night or two to see Tokyo.
You'll be within day-trip distance to Kamakura, too, which is very scenic. The tours people on base can give you directions on how to get there.
Also if you are then during the end of July, sign up for the base's trip to Soma Noma Oi - Yokosuka has one, I know; Atsugi may not. But if you can, go...you have to go at three in the morning and you get back around eleven at night but it's worth it - it's hard to get to on your own if you don't know Japan, but it's a totally fun festival.
Other places that would be fairly easy would be Hakone or Kawaguchiko.
What month are you coming?