How accurate is this website?

[tk21769]

ARWU describes its methodology here:
http://www.shanghairanking.com/ARWU-SUBJECT-Methodology-2014.html

So an undergraduate institution can have a very good physics and mathematics department pedagogically despite not outputting a lot of published research.

Yes.
Conversely, an institution can crank out many research publications, attract top scholars and boatloads of grant money, yet the top scholars who jack up the ARWU rankings may not teach undergraduates at all.