We stayed at an apartment we found on www.budapestweek.com It was listed as the "Aradi" as it is located on Aradi Ut.
The price was $48 for 2 people and $17 extra per person.
It is owned by an individual and an agency rents it for them.
I've done the apartment thing in Vienna as well to great success, found a large 2 bedroom with a great location for $85 night.