Budapest airport transfer options
By public transport:
Take the bus nr. 200 that will take you to Kõbánya-Kispest metro stop, which is the terminal of the blue metro line (M3). From there you need to travel e.g. to Deak ter which is the 10th stop. 2 tickets each (one for the bus and one for the metro). The travelling time to the city centre is 50-60 minutes.
By airport shuttle:
The airport minibus is a door-to-door shuttle service that collects travellers going to the same part of the city and drops them one-by-one. The price depends on the travel distance. Return transfers are cheaper. Their desk is at the arrivals at the airport. A queue is highly probable in peak airport hours.
At the moment unfortunately there is no train appointed for transfers between the airport terminals and the city center. Plans for implementation do exist, hopefully they will be realised soon.
By individual transfer:
Several companies offer door-to-door shuttle service on fixed rates. You can order the service in advance by e-mail or over the phone. The advantage is that they do not drop passengers one-by-one, like the minibus and they pick you up with a nameboard. The price is between usually around 25-30 EUR for 1-4 persons or 35-40 for 5-8 persons.
You can sit into a taxi at the airport, there is one taxi company that resides at the airport, the fare depends on the destination. You can also call a taxi over the phone and ask for an airport fare, this may be cheaper, but not necessarily.
The newest option is Budapest Airport Bus