I want to go Shenyang from Capital Airport of Beijing.

Upon landing, you can take a shuttle bus line 10 from the airport to South Railway Station first.
Then take a bullet train from there to Shenyang. The train takes 4-5 hours on the way.
In addition, yes, you can purchase the tickets by cash. If you wanna book the tickets online in advance, you can ask help for TravelChinaGuide. They are reliable. If interested, their contact information is bookingtravelchinaguide.com.