Personally, I would not recommend visiting any fish market in the morning unless you are really into fish, but somehow getting up early and visiting Tukiji's fish market in the morning is some sort of a trend among tourists. For me is better to walk around there when things are a little bit more calm and you get to enjoy walking around Tokyo's most popular fish market.
Tsujiki is much more than the Fish market. There are lots of interesting spots worth the visit around here. Specially after sunset