They are leasing you your place at market rate. The market rate is affected by supply / demand of rental units, not landlord costs. The only way they could pass the extra tax onto you is if you are not market rate and there is a law that allows them to do this, or if increasing property taxes on apartment buildings decreases the supply of rental units - an unlikely scenario.
Not sure what you mean by this: my landlord can raise my rent for whatever reason (or no reason), by any amount they desire.
I can choose whether or not I want to pay it, of course. But I imagine moderate increases to cover at least some (and maybe all?) of the cost of a property tax increase would be swallowed by most renters, given that moving is also a cost, and often a large one.
In fact, we have empirical evidence for this. The campaign for Prop. 13 claimed that landlords would pass on the savings, but after it was passed, no corresponding drop in rents was observed.