Transportation: 
1.By Railway
Take JR (Japan Railways) Kansai Airport Line from “Kansai Airport” station and get off at “Osaka” station (time required: 1 hour). Change to JR Kobe Line and get off at “Rokkomichi” station (5th station by express train, time required: 25 min./ fare: JPY 1,660).
When taking a train from "Kansai Airport," go up to the second floor of the passenger terminal building and go through the passage to the concourse.
Train Route Finder

2.By Bus
Take the Limousine Bus bound for “Kobe Sannomiya” from Kansai International Airport (time required: 65 min./ fare: JPY 1,800). At “Sannomiya,” go to Hankyu Railways and get onto the Kobe Line concourse bound for “Umeda.” Take the local train to “Rokko” station (3rd stop, time required: 6 min./ fare: JPY 180).
How to use the limousine bus: go out of the passenger terminal building and buy a ticket for "Kobe Sannomiya" at the ticket counter; board the bus at No. 6 bus stop.
KATE Airport Limousine web site

3.By MK "Skygateshuttle" Omnibus
Take MK “Skygateshuttle” to your destination in Chuo-ku, Higashinada-ku, Nada-ku, Hyogo-ku in Kobe city from Kansai International Airport (fare: JPY 2,300, one-way/per person).
Reservation by phone or by web is required up to 2 days prior to the date of use.
When using the MK "Skygateshuttle" omnibus, go to the MK counter on the first floor of the passenger terminal.
MK "Skygateshuttle" web site

4.After arriving at “Rokkomichi” or “Rokko” Station
Take Kobe City Bus No. 16 at bus stop ‘JR Rokkomichi Station’ or ‘Hankyu Rokko’ and get off at bus stop ‘Shindai Kokusai Bunkagakukenkyuka Mae’ (time required: 12 min./ fare: JPY 210).
Kobe City Transportation Bureau