There is a page here at Taiwanderful written by fiLi about the top universities in Taiwan.

National Taiwan University (NTU) is the only university in Taiwan which rates highly internationally. Amongst the other universities in Taiwan, those generally highly regarded include National Cheng Kung University in Tainan, National Sun Yat-sen University in Kaohsiung and National Chiao Tung University in Xinzhu.

I am not sure about rankings for individual subjects.  

Yeah, that's a tough one. I've been trying to figure that out myself for the past year, but there's very little English information on that. The Taiwan Education Ministry and Study in Taiwan websites have some rankings for the technology schools, which is what's hot in Taiwan right now, and although it's not really clear what the rankings are based on (partially because some of it is in Chinese), it's still best to start with those.
If you do figure this out, let me know :P, I've received quite a few emails about this subject.

Just came across a Scott Sommers post about the Taiwanese MoE rankings for the Taiwanese universities. It's a bit old, but it might help a little :

The MOE divides universities in to two categories; General Universties and Universities of Technology and Institutes of Technology (which I will refer to as Technical Universities). General Universities have no formal ranking, although the MOE continues to experiment with various methods of evaluation. Technical Universities have a highly formalized procedure for evaluation and ranking.

The ranking of General Universities has proceeded with caution. For example, last year, the MOE ranked universities based on the publication record of faculty in the Science Citation Index, Social Sciences Citation Index and Engineering Index. The problem with this procedure is that it produced a number of unintended results. Most notable was that the highly regarded National Chengchi University (NCCU) ranked very low on their list. Representatives of the MOE refused to budge under public pressure, stating that this ranking was a work-in-progress and further work was still necessary. It should be noted that NCCU regularly attains impressive status in international comparisons of university quality (1), (2), (3).

The comparison of Technical Universities in Taiwan is much more formalized. Readers with Chinese skills can find more information about this on the Chinese-language site of the Ministry of Education (1), (2). Be forewarned that the Chinese-language site is quite crowded and confusing and therefore difficult to work with.

I believe this is the best reference I found to internal Taiwanese university rankings. I haven't been able to find the update for later years (anyone?).

Good luck.