It is about 1 hour and 20 minutes to 2 hours from Kansai International Airport (both international and domestic terminals) to Katsuragi City by limousine bus and train.
There is no limousine bus stop in Katsuragi city, therefore, you can get off the limousine bus at Kintetsu Takadashi Station (be careful because limousine bus does not stop here at certain time period) or Kintetsu Yamatoyagi Station. Then you can take a train to Katsuragi City from there.
It is about 1 hour and 15 minutes from Osaka International Airport (Itami Airport for domestic flights) by limousine bus and train.
For more details, please refer to the following homepages.
Using Kansai International Airport:
Nara Kotsu Limousine Bus
Using Osaka International Airport:
Osaka International Airport
Osaka Kuko Kotsu Limousine Bus ( bound for Tennoji)