Public transport in Budapest is run by BKV - www.bvk.hu is a useful website and explains how the ticketing system works.
To and from Liszt Ferenc (LF) airport – takes about an hour door-to-door. It’s easy if you are confident using public transport
There’s a BKV ticket office in the airport arrival hall to purchase tickets etc. You will need Forints or a credit card to purchase tickets.
To reach the apartment, take the 200E bus from just outside the terminal building to Kobanya Kispest. Buses are frequent and the journey takes about 25 minutes. Kobanya Kispest is the last bus stop. You must validate your ticket in the machine on the bus unless you have a pass.
From Kobanya Kispest follow the signs for the metro and purchase another ticket or validate another before you go down stairs to the train.
Take the blue line (it’s the only one) for 7 stops to Corvin Negyed. At the top of the escalator follow the signs for the tram numbers 4 and 6 to Szell Kalman Ter. The trams are very frequent. Validate another ticket on the tram if you don’t have a pass. Take the tram (either 4 or 6) 2 stops to Rakoczi Ter.
When you get off, head towards the front of the tram and cross the tram tracks and road to your right. Cross the small road to your left and you have reached the building.
To return to the airport, follow the journey in reverse. When you reach Kobanya Kispest follow the signs for the 200E.
If you are here for 3 days or more you might want to consider buying a pass that will cover you on all the trams, buses, metro – including night buses and your journey to and from the airport. The tickets are valid on the day purchased and for the next 6 days until midnight. In 2017 weekly tickets cost HUF4900. A 3-day ticket costs HUF3500 and is valid for 72 hours from the time of purchase.
They run until about midnight and are a good way of getting around. The number 2 follows the river along the Pest side and is a good sightseeing tram. Catch it from near the Great Market and head in the direction of the Parliament Building. The 4 and 6 take you round the ring road and are really useful.
Trains run until about 11pm.
The bus services are excellent, safe and reliable. They are also really frequent.
If you want to take a cheap trip up the river catch the boat from Boraros Ter (4 stops on the tram heading left from the street). It goes past the Parliament buildings and beyond. Go down the steps from the tram and head a little to your right when you reach the river. You can use normal transport tickets although not on Saturday and Sunday when you must buy a separate ticket.
You can park in front of the building. It’s free between 8pm and 8am and at weekends. Otherwise it’s on a meter (you'll need coins) with a maximum 3-hour stay.
Alternatively there is a 24-hour car park behind the building (to the left) on Bacso Bela Ut.
For somewhere longer term, we have used a car-park at Stadionok at the Sport Arena (a couple of miles away) that is secure and good value. You can use your car during the time it is booked in as they give you a key-card.