Bus from longmen grottoes to Shaolin temple

As the information that I checked, there is no direct buys operating between two scenic spots.
1. When you finish visiting Longmen Grottos, you may take bus no.81 at Baiyuan Bus Stop to Luoyang Bus Stop. Upon arrival, you just need to walk about 100 meters to the Bus Station. Then you can catch a direct bus to Shaolin Temple. This bus operates from 05:30 to 12:30 with the interval of about 30-40 minutes. The ticket fare is CNY19/person.
2. Yes, you can also take a taxi and the fare may be around CNY100 in the day time.

However, shuttling between the two spots in one day with public transportation, your time may be a little tight.

I am afraid no direct coach is available between the grottoes and Shaolin Temple. You may go to Luoyang Bus Station and take a direct bus to the temple, and it is the same for the way back. However, I can see your time is a little bit tight to fully enjoy the two spots in a day by public transportation. Therefore, it is best to have the tour arranged by an agency. If you are interested, I would recommend TravelChinaGuide to you. They will offer good services for you. Here is their contact by the way: traveltravelchinaguide.com.

Have a good trip!