I'll visit to Taipei on January.
Do I need a jacket??

Not really, it's about 20℃.
An umbrella would be needed.

Sometimes it may go as low as the lower 10's, and it will feel colder when it rains. I would suggest you still bring one just in case. Or you can always buy one here! :-)

of course!!! bright one here!

Yes, you should bring one in case.