For Friday night I would recommend Dedesuke in Shimbashi, maybe a ten-minute walk from the Conrad. It's a nice casual izakaya specializing in high-quality charcoal-grilled meats, with some good sake.

Most places in Shimbashi are just closing at that hour, so walking around looking for a place that's open can be frustrating. And there are a lot of touts out on the street trying to lure people into places that are still open, some of them fairly dodgy.

On a Sunday in Tokyo there are tens of thousands of restaurants to choose from, although many sushi restaurants are closed since the fish market is closed.

You're obviously familiar with sushi and kaiseki, but there are dozens of other cuisines worth trying out - this might be a good time to just wander around. If you want something close to your hotel, I'm especially fond of Washoku En on the 42nd floor of Shiodome City Center.