There is a direct bus connection from the Budapest Airport to the city. The line is called 100E.

Buses run in general from very early till very late every 20 minutes, so there is a good chance you can use this way of transportation. For details (with timetable) see the website of the Budapest Public Transport Company (BKK)
For this bus you will need a special ticket which can be bought at ticket vending machines just like other tickets.
On top of this special airport bus ticket you will need one single ticket to take Metro 4 on your way to Bölcső7.
More informartion about public transport tickets
Under normal traffic conditions it will take about 1 hour from the Airport to Bölcső7.
This is how your trip will run:

After getting off Metro 4 at Újbuda központ, this is how you find Bölcső7:



