The fastest transfer from Osaka Kansai International Airport to Kyoto station is the direct Haruka Airport Express operated by JR.
Kansai International Airport
There is only one terminal at Kansai International Airport. After clearing baggage reclaim and customs follow signs to the station. (You will have to go up to level 2 from arrivals at level 1.) You can take the luggage trolley into the station.
Boarding the Train
When you have your ticket, cross the foyer and pass through the ticket gate (the ticket gates are extra wide for you to take luggage carts through). Collect your ticket on the other side and keep it available for inspection on the journey. You will have to remove your luggage from the cart before proceeding down to the escalator platform from here. If you have reserved seats, you can identify the carriage by signs and markings on the platform.
Haruka Express will make a few stops before terminating in Kyoto station. This will be announced with an audio message and bulletin board in the carriage in both English and Japanese. (*About two trains per day do not terminate in Kyoto but carry on to destinations further up the line; in this case be ready to alight a few minutes before arrival in Kyoto.)
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To access Kyoto station from Narita International Airport, take the Narita Express (operated by JR) to Tokyo station and transfer here to the Tokaido Shinkansen (bullet train) to Kyoto station.
Narita International Airport has two terminals, Terminal 1 and Terminal 2. Both have a station where you can catch the JR Narita Express. Follow the signs to the station after clearing baggage reclaim and customs. You can buy tickets for your entire journey to Kyoto at the JR Ticket office in the station. The clerk will help you choose which departure time to take from Tokyo. You can reserve seats on the trains here too. You can use cash or credit card to buy tickets.
You enter the station by passing your ticket through the ticket gate and collecting it on the other side. Your Narita Express Ticket will have a reserved seat number. You can identify which carriage your seat is in by markings on the platform. Keep your ticket accessible at all times because you might be asked to show it on the train. The ticket should be printed in English.
Tokyo Station Transfer
At Tokyo station, follow the signs to the Tokaido Shinkansen for Nagoya, Kyoto, Osaka and Hakata. You will have to use your ticket to pass through separate ticket gates for the Shinkansen. If you have seat reservations for this train too, you can find your carriage number by signs on the platform. (A typical journey to Kyoto from Tokyo stops at Shinagawa or Shin Yokohama (or both) and Nagoya only. It is 36 minutes from Nagoya to Kyoto.) You will be asked to show your ticket once during the journey.
When you get off in Kyoto station head to the exit and pass your tickets through the ticket gate. The tickets will not be returned to you this time.
For additional information