No，You should go to Shanghai Huangpu Bus Station to buy the ticket in advance.
It is hard to get a seat from Shanghai to Hangzhou airport.
I advise you to take D-train from Shanghai South Railway Station to Hangzhou Railway Station. Then take the airport shuttle bus to the airport.

You mean even if I get to buy the bus ticket, I may not get a seat in the bus as well?

Train tickets also need to buy in advance right? Can I buy tickets departing from Shanghai to Hangzhou in Hangzhou train station or any other station? Or it is necessary only buy from Shanghai Train Station.

Also, my last destination before I depart to Hangzhou maybe in Shanghai Old Street/ Taobaocheng Market. Should my train departure be from Shanghai South or Shanghai?

Yes, you'd better buy ticket in advance at the bus station, tickets withing 3 days are available there. If you buy tickets, you'll surely get a seat.
Train ticket also need to buy in advance, you can buy ticket of Shanghai to hangzhou at Hangzhou Railway Station, it's available. There are trains from Shanghai South Station and Shanghai Station, but there are more train at Shanghai South station.

Thanks Mr Ross and Ms Supergirl.
I'll take note. Thanks for all the guidance.