The Xining airport bus is also leaving from the west side of Xining. It leaves from the entrance of the Super Leisure Hotel, just East of the ShiDa main gate at WusiXiLu. It leaves at irregular intervals, so you need to go there first to find out when a bus leaves. You tell them your flight number and departure time, and they will tell you which bus to take. You can also phone: 13997030201 (Chinese speaking).
The bus/van will take you to the main airport bus departure location on Bayilu, the other/east side of Xining, and then you have to transfer to the main airport bus. Price 21 rmb in total.
Super Leisure Hotel, 舒泊来雁酒店
Wusixilu nr 36, 五四西路 36号
The airport bus stop is at the most left/west corner of this yellow building on Wusixilu.
Map of Airport bus West side departure location.