Taipei is slightly cheaper than Singapore and is more traditional Chinese - I think that is definately a better alternative if it's included in your airfare. Just don't get caught at the top of the Taipei 101 building in an earthquake!:D
As long as you do not drink I think you should be okay money wise. Tokyo I am the least sure about however, because that is the one place on your list I haven't been but I have heard a lot about how pricey it is. As long as you stick to seeing sites that are inexpensive, use metros and walk a lot, no problem. As expensive as HK is, for example, the Star Ferry
that runs between HK Island and Kowloon is very cheap. Just wandering the streets and checking things out is a great way to spend the day there.
Most Asian countries have a lot of street vendors selling assorted food - don't be afraid to eat the street food - its generally good and very cheap, but check out the vendors and go to the ones that are busiest.
Save your shopping for Thailand, its definately the cheapest place for all sorts of clothes and other handcrafts.
If this is your first trip to Asia it will be an eye opening and wonderful trip!!