in Singapore, while you may be too busy with the zoos and bird park to visit any of the less manicured places, I would highly recommend you do try the Botanic Gardens. They open at 5am (though the sun doesn't come up till about 7) and you can easily fit in a couple of hours there in the morning before the zoos open, or in the evening. Doing this will give you a head start on Asian birding without taking away from zoo-time, and you can see how you go with spotting and identifying all the common species there. There is also a "rainforest" area in the Gardens which will give you a feel for what its like birding in a real forest.
Even if you don't go over to Pulau Ubin, a trip to Changi Village just to see the Goffin's cockatoos and moustached parrots would be good for you. There are also lesser sulphur-crested cockatoos in Singapore, although their numbers are declining and they are getting harder to find (Sentosa Island has them although I've only seen Goffin's there; the big sprawling expanse that is the Bukit Brown cemetery is also supposed to be a good spot for them). Little corellas have more recently become established but there aren't many so you probably won't see any. Long-tailed parrots are native, and can be easily found in the Botanic Gardens (but hard to get good views of because they are quite flighty).
If you did get some time a walk up Bukit Timah would give you an idea of what hill-climbing in the tropics will be like (hint: hot and sweaty!), and because there's only one tree shrew in Singapore (Tupaia glis) as opposed to the ten in Borneo, at least you'll know what you're looking at!
