From a Hungarian colleague:
Resto:
Paprika Dozsa Gyorgy ut 72, Budapest 1071, +36 20-294-79-44
Must book!
Cafes and bars: Vaci Street in Pest, most famous is
Gerbaud In Buda, lots around the castle.
More insider information is being sought from her brother, who is on the spot (Pest or Buda, one of the two

).
EDIT: this from the brother:
Szegedi Halászcsárda - Budapest, Belgrád rakpart