Look here
http://www.dca.ca.gov/publications/landlordbook/problems.shtml
and refer to your rental agreement.
I suspect that, since a rental unit must have a functioning bath to be "habitable", it is the landlord's responsibility to make the drain function unless the problem is due to the tenant's "neglect" or "abuse". Normal bathing is not neglect or abuse, no matter how hairy you are.
You might also do some walking through the yellow pages and find the local agency that handles rental matters, if you want to be 100% sure of the answer.
Personally, I would absolutely
not pay the bill. In all the years I rented in CA, I never had to pay for drain declogging or similar fix-it stuff, admittedly I usually fixed it myself but sometimes I just had to call the property manager. If they gave me a hard time, I would find another place and give notice. I'm assuming there is nothing uniquely desirable about your rental unit.