Kobe to Kyoto by train?

Is there a way (preferably by bullet train) to get from Kobe to Kyoto without a transfer?

Yes, JR connects the two cities directly by both rapid trains and bullet trains (also known as shinkansen). You can find all the details here:
http://www.japan-guide.com/e/e3554.html?aFROM=2158_2159

Yes, You can do this on the shinkansen or via JR special rapid train. There is only 1 shinkansen per hour that falls under the JR pass and takes 30 minutes.

The shinkansen station in Kobe is not centrally located so using a JR train might be more beneficial.

If you don't mind changing shinkansen trains at Shin-Osaka Station, you should have 2-3 shinkansen connections per hour that are covered by the Japan Rail Pass. The Nozomi and Mizuho trains are not covered.