Can I purchase a Bus Transport card at Hangzhou International Airport?

The airport doesn't have service counter that sells transportation card.
So you can take airport shuttle bus to go to city center first. You can buy the card in almost every subway station. Or there are also many special service counter that provide this service. Let me know your location in town and I can find a nearby site to get the card.

I will be staying at the Zijingang Campus of Zhejiang University. Is there a nearby site to get the Transport Card?

You can buy the card in Agricultural Bank of China. There is branch office located inside your campus.