You can take the hotel shuttle bus from the Westin Osaka to JR Osaka Station. (Free of charge). From there, you can catch a direct train to Kyoto. It takes about 30 mins.

At Kyoto station there's a shuttle bus to the Westin Kyoto. Make sure you check the hotel web site to confirm the operation times - It stops running fairly early. It leaves from the South side of the station, which was a building site when I last visited. That made it very difficult to find the right bus stop, so I would advise you to ask early and often for directions.