There's an area around a canal in central Kyoto (Kiyamachi) that's one of my favourite places for going out ever.

The are maybe 20-30 bars with just ~8 chairs in each, with some really unique styling. It's like Berlin, but with Bonn-like attention to detail in the creation of cocktails.

Pretty Space Mushroom Bar is unique, but my favourite has to be Dog Bar - a small bar with a yapping dog sitting on the bar. Genius.

I didn't do the underground Tokyo scene (which sounds worthwhile), but for anyone who has seen Lost in Translation and those who haven't I can recommend a drink or two at the New York bar at the Park Hyatt. Annoyingly they charge a 2,000 yen cover charge even for premium Hyatt members staying at the hotel, though.

Edited: As noted 2 posts below, I was being over-optimistic about the amount that I spent that night, so entry to the New York bar has been corrected...