Use the official taxi partner of the airport: Főtaxi.
You can order a taxi through their website:
Főtaxi website – English version
You can either call them or book online through their website.
Paying cash or with credit card is possible. You need to indicate your preferred payment method at the time of booking.
Travel time will vary with traffic conditions, so I suggest to check the estimated travel time with Google Maps. Spoiler: if you calculate 1 hour fro the trip, you will be on the safe side under normal conditions. Of course if there is a diplomatic visit and the roads are closed, or there is a major accident, that is a different case.
