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If you do not want to pay for two seperate transport rides (350 HUF each ride/ticket), you can buy a Transfer Ticket (530 HUF) valid for two rides between metro trains, buses, trams and so on.
Your first trip is valid for 100 minutes and your second trip (the transfer) is valid for 80 minutes, but your transfer must be taken within the first 100 minutes. See the BKK Transport T&Cs.
This 24 Hour travel pass is valid on the metro, tram, trolley bus, bus and boat in and around Budapest city centre.Daily Travel Pass
It is ideal for visiting residential areas as well as tourist areas.
Budapest now has an Airport Shuttle Bus service, the 100E, that travels from Budapest International Airport to Budapest city centre; and back. The exact route, which should take 30-40 Minutes, is: Liszt Ferenc Airport (Terminal 2) - Kálvin Tér (Metro station) - Astoria (Metro station) - Deák Ferenc Tér (Metro station). The stops at Kálvin Tér and Astoria are to drop passengers off only.
The first bus leaves Budapest airport at 5am, with other buses leaving every 20 Minutes from then. The last bus leaves Budapest airport at 1:20am (the following morning). The return journey is: Deák Ferenc Tér (Metro station) - Liszt Ferenc Airport (Terminal 2). It is also expected to take 30-40 Minutes. The first bus from the city (Deák Ferenc Tér metro station) leaves at 3:40am and the last bus from the city leaves at 12:40am (the following morning).
A single, one-way, ticket currently costs 900 HUF (around £2.46) and can be bought from a BKK Ticket Machine (Card or Cash), A BKK Customer Service Centre (Card or Cash) or On-Board The Bus itself (exact change needed).
NOTE: The Kalvin Tér bus stop is actually on the main road called Üllői Út.
NOTE: With the return journey from Deák Ferenc Tér (Metro station) to Liszt Ferenc Airport (Terminal 2), the first two buses (4am and 4:30am) stop at the Astoria metro station for boarding purposes only; and at the Kalvin Tér metro station as standard for boarding purposes only.