Like always, I am sort of biased. The "Taxi Experience" upon arrival is part of the welcome "home" experience.
But first, the bus:
The 100 bus will cost a couple 1800 forints. Depending on where you are staying you may need to connect to the metro or tram system for another 1060 forints for transfer tickets; so 2860 forint total. That's about $11.50.
According to the BKK website the bus takes 40 to 60 minutes and it leaves every 30 minutes.
So, lets say you show up 5 minutes after the bus left. So you have a 25 minute wait, then a 50 minute ride.
Then if I were going home i would have to walk to the M1 line for that figure maybe 5 minutes if you don't know exactly where you are going. At night the M1 runs every 5 to 10 minutes depending on the time in the evening. Lets say 5 minutes. My stop is the Opera stop which takes 5 minutes. Then i have a 3 minute walk home. So the bus will take you maybe 1 hour 33 minutes. I would figure 1:15 to 1:45.
Now the Taxi
There is a black line on the floor in baggage claim that will take you through the arrivals hall and out to the FoTaxi (airport contract taxi company) kiosk just outside the doors. At the kiosk a polite young person will ask you where you are going (in perfect English). Then the person will give you a receipt with the taxi number and the maximum you can be charged. The receipt also lists the location you are going. If there is any confusion it is handled at the kiosk and not in the taxi. Next you walk about 40 feet to the curb where another polite young person will help you get in the right taxi and help you with your luggage. The whole process takes something less than 5 minutes. You hand the receipt to the driver and you are off. My ride is generally about 35 minutes and stops at my front door. So maybe 40 minutes. The fare with tip runs about 7000 forints or $28.00
So yes, the bus will save you $16.50!!
Just be very careful, if you eat one dinner in District V or the lower ends of VI, VII and VIII or you will instantly loose that $16.50 you saved by loosing half an hour and standing on a crowded bus.