You don't hve a ton of time, and while I am no expert by any means (only been in Japan once - over two years ago), I did stay in the same hotel for a few days. The view from the bar at the top of the hotel is fantastic.

Wandering around Roppongi at night is fine, I certainly did that and met a load of people, expat kids, expats, local japanese, you name it. Nothing really spectacular, but fun nonetheless.

I would also try and wander the area. Take the underpass and cross the street, and just wander around the area by walking onto smaller streets/pathways, poking your head into every shop/business possible, just to see what they are. Wander into a tonkatsu restaurant or a random ramen house and if you get a good feeling, sit down and eat. Take some photos, memorize 10 Japanese phrases (excuse me, hello, thank you etc), use them in the above settings and smile. Take photos (if you like that kind of thing) That should set you.

You could do things like the Imperial Gardens and all, but it does not seem like you have the time. Enjoy!