Taipei: Jan 1 is a holiday but in practice everything is open. Recently businesses tend to stay open even throughout the Chinese New Year. The Taipei 101 fireworks are impressive but it can get crowded if you want to see them up close, fortunately there is a good choice of viewing spots farther afield. Weather-wise, mid-December to mid-February is not the most pleasant time of the year to spend in northern Taiwan: it can feel much colder than the temperatures would suggest. There is not a lot of rain though (if at all, it's just drizzle).

Singapore: not sure what they do for the New Year, it's not an obvious New Year destination (not that Taipei is). Weather-wise, it's the middle of the rainy season there. Temperatures are similar throughout the year.

So far Taipei seems to be doing much better as far as the virus is concerned: practically everything remains open. Might still be more difficult to get in at the time though.