October 1st at 2:53pm
Let’s consider the cheapest (?) option, namely getting in by public transportation.
I only recommend it to the ones who really love adventures. This might take up to 2 hours for newcomers and the old subway cars don’t have air-condition so summer heats can be very demanding. In details you pretty much have to exit the terminal, get on bus number 200E that runs every 7-8 minutes from 3:30 am to 10:59 pm. Departure times after 10:59 pm: 11:21, 11:36, 11:51, 00:21, 00:59, 01:29, 02:29 and 03:29. If you are taking a bus that runs between 3:30 am and 10:59 pm, then you can transfer to the subway line 3 at ( Kőbánya-Kispest ) and just come to the city center. In case you are taking the one that runs between 11:00 and 00:21 you have to transfer at Határ út and get on a night bus (914, 914A, 950, 950A), which needs some precaution. If you happen to take the one between 00:22 and 03:29, you should transfer at Bajcsy-Zsilinszky út again to 950 or 950A night bus. Such a journey requires the same amount of single tickets as the number of vehicle you use. In 2015-September on ticket costs 350 HUF. (approximately 1.2 Euros). Please also keep in mind that the tickets needs to be validated before using the bus or subway. The tickets can be purchased at the booths upon exciting the terminal.
The second option is to use the AirportShuttle-Minibus service.
It provides a really convenient and reasonable airport door-to-door transfer between the Airport and any point of Budapest and its fleet operates 24 hours a day. The transfer operates as a shared-ride service, which means that passengers travelling to /from the same/ closely situated address are escorted in the same vehicle. The travelling time is optimized and supervised by a GPS-based fleet-monitoring system, which is capable of displaying the actual position of all vehicles in real-time. Every vehicle is equipped with dual-air conditioning systems and free Wi-Fi hotspots for gratis internet access, further increasing the passenger comfort during the journey. The price for one way varies from 1900 HUF to 6900 HUF (6-22 Euros) depending on the distance. I would also mention that you might need to wait until the bus fills up with passengers.
The third option is to come by cab.
I would strongly discourage everybody from arbitrarily taking cabs while in Hungary. Rip-offs happen almost everyday, meaning that you cannot get out of the taxi unless you pay couple of hundreds of Euros. Alright, it might sound the most convenient way to travel in, but it can also become quiet costly. Starting September 1, 2013 the system of taxi regulation is in effect in Budapest; all taxis must be painted yellow, all must have a standardized meter and rates, all must have a lighted TAXI sign on their roof to show when they are available. They are no longer able to charge flat rates and always must work off the meter with a standard fare of 450 Forints to start and 280 per additional km of driving, 70 per waiting time minute. An average ride would cost about 27 Euros and Sometimes it happens that the driver doesn't take you to the house, but drops you off at a nearby crossing. English speaking well educated taxi-drivers are also pretty rare. In case you are still considering this option, you can find the offical partner of the airport “Főtaxi” at the exit of the terminal and request a cab there.