The York plasma Institute is less than two miles outside the city of York. See a walking route in google maps here.
6, 44, 66, 45, 46, X46 & X47 all travel from York city centre to the University of York. The 66 is the most convenient and regular bus and the bus stop on Piccadilly is convenient for many hotels in the City Centre. You can buy a bus ticket using cash on the bus.