There are countless guidebooks and online guides to Tokyo, which offers a lifetime's worth of attractions. With so little time, it will be hard to choose, but only you know what interests you.

Shinagawa is, as others have said, boring. but it is the most convenient area for transferring from the N'EX to Haneda. Check into your hotel, and then return to Shinagawa station to ride the Yamanote Line (silver train with green stripes) to Shibuya or Shinjuku for restaurants, clubs, and just plain walking around. Since you're just in town for a brief time, buy a single train tickets from the vending machines, put it through the turnstile to be stamped, and keep it till you reach your destination, where you will once again insert it in the turnstile. This time it will not be returned. (Get some cash at NRT.)

Before you set out on foot, spot a landmark and keep it in sight as you wander, since both neighborhoods can be confusing. (I am an Old Tokyo Hand, but I am capable of getting lost in Shibuya.) If all else fails, ask a younger person for directions to the station (they are more likely than older people to remember their high school English).

For a nighttime view of central Tokyo, ride the roughly circular Yamanote Line all the way around to return to Shinagawa.