I'll stay at Green Court Serviced Apartment. Where can I buy tickets/board Shanghai sightseeing bus?

I presume you refer to the branch on West Beijing Road. Well, you should walk to Urban Planning Exhibition Center which is close to People's Square subway station within 10min, then take city sightseeing bus line 1 or line 2.

There is a ticket booth near the bus stop where you can get the ticket. Or just board and buy a ticket from the driver.