1 hr to get into town by bus(200)/metro(m3). Don't know about taxis, but I would imagine it would not be much quicker.

I would do one of these two itins:

0. Get a day ticket from the machine at the airport. A single ticket is only good for one trip. Even if you're transferring on the metro, you're required to have a separate ticket for each line. A day ticket saves you $$ and the hassle of having to stamp/buy multiple tix. Validate the ticket on the bus (or maybe you have to manually write in the date, I don't remember)

Itin 1:
1. Take bus 200 and M3 to Kalvin Ter and walk to the Great Market Hall. Get a quick bite to eat from the food stands upstairs.
2. Walk down Vaci Utca, passying Vorosmatrty Ter
3. Make a left at the chain bridge. Cross it and take the funicular (siklo) up to the castle grounds.
4. Walk down the back side of the hill to Moszkva Ter.
5a. Get on the metro (M2), take it to Deak Ferenc Ter, transfer to M3 and go back to the airport.
5b. Alternatively take one of the combino trams (line 4 or 6) to Kalvin Ter and go back to the airport
5c. Do 5b. but get off at Oktogon and walk to Deak Ferenc Ter or take the millenium subway (M1) there and transfer to M2. Go back to the airport.

Itin 2:
1-2. Do #1-2 from Itin 1.
2. Make a right at Vorosmarty Ter and walk to Deak Ferenc Ter.
3. Walk down Andrassy Utcz all the way to Varosliget Park. (the walk is quite long, so watch your time here).
4. Take the Millenium Subway (M1) to Deak Ferenc Ter and transfer to M3, go back to the airport.