The shuttle service is still quick and easy - much cheaper than a taxi with little to any more time required. You can buy a ticker while you wait for your luggage at baggage claim, then you check in and wait 5-10 mins in the arrivals hall.
If you really want to take a taxi, I would ask someone in Budapest for the number of the taxis they use -even your hotel. Taxis in Budapest can vary wildly in price and quality of service, so I would call one to meet you at the airport rather than just catch one that is waiting there. over 6 years ago