This is how to get your hotel in Osaka from KIX

You can travel from KIX to Osaka by train or bus.
In the case of trains, using the Limited Express Haruka will save time and reduce the number of transfers on the way. 

NOTE; Haruka stops at Tennoji, Shin-Osaka, and Kyoto, so it doesn't stop at Osaka Station. 
For details, please see Access on the KIX website or search for Limited Express Haruka. 

In the case of buses, there are 3 bus companies operating (HK, HS, KK) from KIX to Terminal 2 and then to Terminal 1 in that order. 

※HK stands for Hankyu Bus, HS stands for Hanshi Bus, KK stands for Kansai Kuko Bus

You can get off at major hotels in Osaka city by the end of Senri-Chuo. 
Hotel stopover: Hilton, New Hankyu Hotel, Hotel Hankyu Respire OSAKA, Hotel Hankyu International, Westin Hotel and HERBIS(Not hotel). 

Please see the attached bus schedule below. (written in Japanese) 

NOTE; The time may change due to the circumstances of the bus company