If you are a foodie, this place would no stranger to you. I actually don't have anything specific to recommend to you but just some thoughts that I would like to share.
The fish market is divided into 2 blocks. The famous one is the "wet" section, where fish auctions are held in the wee hours of the morning. Within this block, there is a cluster of small sushi bars and other eats as well. If you do a 'google' on the best sushi in Tsukiji, most likely, Sushi Daiwa would come up. This tiny sushi bar is located somewhere within the cluster of restaurants here. But mind you, be prepared for a loooong wait. I was there on a Saturday morning, 8am (!) and was told I had to wait for about 4 hours before I could sink my teeth into fresh sushi. I say, no thank you!
Just 2 doors down, I waited for about 15 minutes before I had a delicious don topped with uber fresh uni, negitoro and blood red and tender maguro, all just for Yen 1,350. That included a bowl of miso soup and pickles too. Strangely, there was no photography allowed so you would just have to imagine.
So the point is, you are in the middle of the busiest fish ports in the world, how bad can your fresh seafood meal get? Give other restaurants a chance.
The other block, which is located just 1 street away, is more like our Singapore typical wet market - there are many stalls selling different things - meats, seafood, omelettes, knives, bonito flakes, fruit and vegetables, etc. You get the gist. There are also sushi bars and non-discript restaurants, like the one picture above, scattered all along the pulse of the market. It was almost 1pm when I ordered my kaisen don for Yen 1,000. It was delicious.
Tsukiji market starts closing by about 1pm. I don't think they are opened on Sundays as well as occasional Wednesdays. So you have to do your homework before visiting, lest being disappointed.