Take Metro Line 9 from Beijing West Railway Station, pass 4 stops and arrive at National Library Station.
Take Daxing Line of Metro Line 4, pass 2 stops and arrive at Xizhimen Station.
Take Metro Line 2, pass 5 stops and arrive at Dongzhimen Station.
Walk about 5 10 meters to Dongzhimen Station.
Take the airport line, pass 2 stops and arrive at T3 terminal.
Walk about 280 meters to Beijing Capital International Airport.
Or you can go this way.
Walk north from the North Square of Beijing West Railway Station for about 10 minutes, and you will see the Military Museum Station of Metro 1 Line. After getting off the subway, you can transfer to Metro Line 2 at Jianguomen or Fuxingmen Station and transfer to Dongzhimen Airport Express Rail. You can also take the bus directly to Guomao Station and transfer to 10 line to Sanyuanqiao Airport Express Rail.
Have a nice trip!