You need to go to Osaka first, then please see "direction from Osaka".
From Osaka by trains http://seasidekyoto.blogspot.com/2013/01/how-to-get-to-amanohashidate-from-osaka.html
From Osaka by direct bus http://seasidekyoto.blogspot.com/2013/01/direct-bus-from-osaka.html
By airport shuttle bus from Kansai airport to Osaka
Duration: about 1hour
Bus stop #5
Fee:1500yen/single adult
Time table http://www.kate.co.jp/pc/time_table/osaka_sta_tt.html#from
Bus map http://www.kate.co.jp/pc/e_time_table/e_map/e_osaka_map_n.html
Get off at Hotel New Hankyu(新阪急ホテル).
By direct train(Kankuu kaisoku(関空快速)) from Kansai airport to Osaka
Duration: about 1 hour
Fee:1160yen/ single adult
Time table http://www.hyperdia.com/en/
Please type, From"Kansai airport", to "Osaka" then you can find the name of train "JR Kansai airport rapid service".