Hi azrahelle,
12 and a half hours is quite a long time, it's definitely worth going into town. I would have loved to show you around but I will be off enjoying my own well-deserved holiday at that time. However, if I were you, I would probably visit the castle district at Buda, which is one of the most favoured tourist sights; what's more it offers a splendid view over Budapest. You'll also find there a number of museums (I personally like the Budapest museum the most), the royal palace, the St Matthias church (the prettiest in the whole country). I wouldn't miss the castle labyrinth either:
http://budapest.forumactif.com/viewtopic.forum?t=12 .
To get to the castle you can take the airport minibus to Moszkva tér (a huge square at the Buda side of Budapest). You can also take a cab (but don't take the ones queuing at the airport rather call one of the bigger companies listed in this thread:
http://budapest.forumactif.com/viewtopic.forum?t=5 the taxis at the airport will charge you two or three times the normal price), or you can use public transport as well, see here:
http://www.bud.hu/english/transportDon't forget to pick up a free tourist map at the airport, they are normally easy to find.
Once you are at the Moszkva tér take the castle bus up to the castle district. (Ask somebody for the várbusz). There is a metro station at the Moszkva tér, you can buy bus tickets there.
In fact the airport minibus might take you up to the castle district too, I don't think they are allowed to enter because it's a restricted area but it might put you down at a walking distance. Ask at the airport.
It is very easy to spend a whole day in the castle district but if you have time you could walk or take the funicular down to the Lánchíd (Chainbridge), cross the bridge and walk down south next to the Danube this way you can take a look at the castle district from a distance too. You can also take the street parallel to the Danube, which is the Váci utca (starts a little further down than the bridge), a shopping street for the tourists (so you can imagine the prices

). The blue metro is at the end of the shopping street, you have to walk some hundred metres up along the big crossroad. To get back to the airport you can take the metro to Köbánya-Kispest (which is the end station) and from there the bus 200 which stops both at terminal 1 and 2.
Whishing you fun in Budapest

Alice
PS. You could also consult the tourist sites mentioned here:
http://budapest.forumactif.com/viewtopic.forum?t=5
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I wonder if I shall fall right through the earth!
How funny it’ll seem to come out among the people
that walk with their heads downwards!