Take the airport bus, 25 yuan per person. After getting on the bus, ask the driver where it is near to get off at Renmin University (University for Nationalities). I remember that the route taken by the airport bus used to pass through the People's Hospital. Now the new route is to go to the city center first and then go back to Bayi Road where the People's Hospital is located, so it is safer to ask the driver.
If you take a taxi at the airport, choose the one that has just sent the guests to the airport and returned to the city immediately. Basically, it's 100 yuan, and you can bargain (the driver will definitely be willing to pull you, otherwise the driver will not make money if you go back to Xining empty).
It's about 30 kilometers from the airport to the People's Hospital by taxi.