The Rényi Institute is right in the city centre, close to the underground stops Astoria and Kálvin tér (see the map below).
Upon arrival at the airport, you can take the Airport shuttle bus 100E. It comes directly to the centre, its first stop is at Kálvin tér, the second stop is Astoria.
The Airport shuttle bus single ticket is available for purchase at the airport and all BKK customer service centres, ticket offices and ticket vending machines.
For your stay in Budapest we suggest two hotels, both very close to the institute:
Danubius Hotel Erzsébet (H-1053 Budapest, Károlyi utca 11-15.)
Danubius Hotel Astoria (H-1053 Budapest, Kossuth Lajos utca 19-21.)