How to get into Kyoto
Taking the train is cheapest way. From Osaka Airport station, take the Monorail to Hotarugaike. Next take the Hankyu Takarazuka Line (private line) to Juso station. Then take the Hankyu Kyoto Line to Karasuma Station (650, 1 hour). This is in central Kyoto and quick ride on the Karasuma Line (subway) takes you to Kyoto station.
An easier way is to take the direct Osaka Airport Limousine Bus (リムジンバス伊丹空港線) to Kyoto station (1280 yen, 1 hour).
How to get into Osaka
From Osaka Airport station, take the Monorail to Hotarugaike, then take the Hankyu Takarazuka Line (private line) to Umeda (420, 22 mins).
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