Just got back a few weeks ago.
-Tsukiji Fish Market: If you want to see the famous Tuna auction, do not be there a minute past 3:30am. I woke up because I was jet lagged and figured what the hell.... Got there at 3:40am and it was sold out. Got in line for the famous Sushi Dai at 4am, opens at 5:30am and where I was in line was a 4 hour wait. A local told me to go next door to Daiwa Sushi as it is just as good and half the wait, it was amazing. Easily the best sushi I have had in my 33 years.
- I took a 6 hour bike tour with Tokyo Bike Tours. Awesome way to see Tokyo and experience areas you would otherwise never find.
- Akihabara: this place is just wild. It is the Mecca for anyone who likes robots, retro video games, Anime and girls dressed up like maids. You have to go see this place as it is culturally so different than anything I have seen in Singapore, HK, Bejing etc
- note on accommodations, if you need somewhere to stay consider AirBnB. You can get a nice private apartment that is spacious for Tokyo at a good price compared to hotels.
I'll write more later but I'll just say that Tokyo is nothing like the other Asian cities you mentioned, especially Singapore. Tokyo has the most unique culture I have experienced anywhere.