Sorry, TRK, I failed to read the address properly: I don't know who told you that Szekesfehervar is 'on the outskirts' of Budapest - it is some distance from Budapest - round about 40 miles.
At that time of the morning going by train is probably the best option - go to the MAV site to check: the journey takes about 75 minutes (trains outside Bpest go at a crawl). There is a train at 06:41 which gets there around 8:00: surely your hosts can pick you up from the station ? Szekesfehervar isn't a particularly big place (my wife's cousin lives there, know it well) so it will be a 10 minute drive tops.
A minor problem is that the train goes from Deli station, which is in Buda - in fact you can walk to it down the hill from the Var in about 15-20 minutes. If you do a google search on 'Deli Palyudvar hotel' it brings up some hotels, the Mercure Buda is a short distance away.
So your plan might be,
get a taxi from outside the airport * to your hotel,
walk to Deli and buy your ticket for the next day (+ return?)
and get some BKV tickets for the metro/ bus etc,( book of 10)**
get the metro from Deli PU to Deak Ter and change for Vorosmarty Ter
wander round the centre for a while, including eats
get the metro back to Moskva Ter (they recently changed the name to Szell Kalman Ter, so look for either name)
walk out of the metro station, there are some steps up leading away from the trams and metro entrance, at the top of those steps you will find a bus which will take you up the hill into the Var - get off outside the Hilton.
* walk out of the airport terminal, there is a Fo taxi stand in front of you - give them your address and off you go, it is a fixed fee.
** each time you make a journey on a bus, tram or metro you have to stamp a ticket - if you buy 10 you will have some left over, but is saves faffing, and you can always hide it in expenses (if MPs can do it, why not the rest of us ?).
How do you plan to get back to the airport ? care nees to be taken with taxis if not pre-booked.
There are ATMs at the airport if you need one, plus all over town.
Edited: 4:50 am, November 21, 2012