Unfortunately, there is no direct transportation to and from KIX. Conrad Osaka is connected to Higobashi Station of Yotsubashi line. As a local I would take a limousine bus for Umeda, get off at Herbis Osaka bus stop and then take a Yotsubashi subway line for Higobashi, one stop from Umeda, or take a limousine bus for Umeda, get off at ShinHankyu Hotel and catch a taxi to the hotel. If you are familiar with Osaka subway line, go to Namba station by Nankai line and transfer to Namba station of Yotsubashi subway line and get off at Higobashi station.