I think you could take a train to Zhengzhou from Beijing firstly and then take a direct coach from Zhengzhou to the Shaolin Temple.
Train from Beijing to Zhengzhou, so many, search one for you at https://www.travelchinaguide.com/china-trains/

From Zhengzhou to Shaolinsi, out of the Zhengzhou Railway Station, there is a coach station where buses to Shaolinsi depart frequently. The journey is about 1.5 hours and the ticket fare should be 25yuan per person~