Official instructions for access to RIMS.
The closest airports are KIX (Osaka Kansai international airport) and ITM (Osaka Itami airport). If you come from abroad, flying to KIX and then taking a train (e.g., JR Haruka to Kyoto station followed by a taxi or common transportation) or a door-to-door shuttle (e.g., MK taxi) are standard solutions.
Some people prefer to fly to/from Tokyo HND (Haneda) / NRT (Narita) as there are more choices of flights. Flying to NGO (Nagoya Chubu Centrair airport) is a good option too. Although not obvious at first sight, it does not take much longer to reach Kyoto if you land at a Tokyo airport in the daytime, thanks to the Japanese bullet train (Shinkansen).
In general, train tickets do not need to be reserved in advance in Japan. On the other hand, shuttles must be reserved ahead of time, and this can be done online.
Either way, if you land in the evening, plan the domestic part of your trip ahead of time (e.g., if you are not in a standard hotel, confirm the latest check-in time, and if you do not travel with a shuttle, check the schedule of the last train of the day — feel free to talk with the local organizers about your situation if needs be).