What other mode of transport available from Taoyuan to Tainan beside the high speed train?

Well, except high-speed train, you can take Taiwan Railway; but it is slower than the high speed.
At the airport, take airport shuttle bus to get to Taoyuan Railway Station. It takes around 1.5 hours on the way. And the ticket fare is NT$59 per person.
Then you can take the train to get to Tainan Area directly.
The schedules are from 06:00 to 19:51 with the duration of around 5 hours. The ticket fare is NT$ 492 to NT$ 637.
After arriving, take a taxi to go to the university.

Hi Carrine, Thanks so much for your help. Very informative & clear direction. I will arrived in Taoyuan airport around midnight, just wonder whether the airport shuttle bus still operate at such time! May I know how much it will cost to take a cab/taxi from the airport to Taoyuan Railway Station? Thanks!

Yes, there still have airport shuttle buses at that time. So don't worry.
Taking taxi is a much expensive transportation for you. Generally speaking, it's not suggested to take a cab unless you have emergencies.