"Cool places" can carry many different meanings. If I were to make a suggestion, I would seek out some of the different yokochos (back alley collections of bars, etc) and just poke your head in. I've done this a bunch (most recently last week) and have never been treated badly. Most nights I would end up the only english speaking person in the place. Granted, most of these only seat 5-10 people. For me that's a win.

For a more formal cocktail experience, I can heartily recommend Bar Gen Yamamoto. It's a tasting menu experience of 4-6 cocktails that's outstanding. Reservations required

Yukari is actually a friend of mine. She tends to focus more on small group tours but could likely offer some other suggestions. If you would like her contact info it's on her website, http://foodsaketokyo.com

The city is well served by just roaming and being open to what happens!