The airport is linked to the city center by train in 2009. The
trainride take about half an hour. But at the airport train depart and arrive near
Terminal 1. (the old
terminal, which was renovated to accomadate the
low cost airlines).
In 2009 there is no railway near the new Terminal 2 which is at the
other side of the runway from Terminal 1. The only way to get to the
railway station from terminal 1 is first take a regular
busservice, linking the two terminals. This bus service also goes from
the airport to the Budapest metro. Many people take this bus to reach
the metro, instead of going to terminal 1 and then walking to the
trainstation.
Taking the railway in Hungary is very cheap in 2009. The trainride will
cost less than 1 euro. But when you take the train in the city center,
you will probably best ask a local at which trainstation you need to
get off for the airport, as there probably no announcements on the
train. And often you get the wrong advice.
Taking the bus and metro might be a better option for foreigners.