Restaurants in Budapest
There is almost no kosher food in Budapest other than the lunchhall at the Orthodox synagogue (back into its courtyard as I recall), but there is excellent Hungarian Jewish style cooking at Rosenstein and at Fulemule (owned by the people who own the Hotel Anna).
More broadly, Fatal is a tourist place in the middle of the hopelessly kitschy tourist street Vaci utca, but those are not necessarily liabilities. Menza is adored by tourists and locals alike for no reason I can discern, though it is a pleasant place to sit and the food is ok.
For wine by the glass, the best places in town are Klassz on Andrassy ut and Borbirosag behind the Central Market. For cheap eats, the Kadar Etkezde is hard to beat, on Klauzal ter, but there are a growing number of lunch spots with 3 course menus for under $6 -- notably Ruben on Magyar utca and Stand Bisztro at Sas utca.
I was only in Budapest for a little over a day, but had some solid, reasonably priced meals at Fatal (Vaci utca, 67, District V, Budapest, Hungary Tel: +36 1 266 2607), very close to the not-to-be-missed market and Menza (VI. Liszt Ferenc tér 2 +36 1 413 1482) Both serve Hungarian fare (gulyas, meat dishes with hungarian noodles, etc.), with Menza being a bit more upscale with some wiggle room to try different Hungarian wines by the glass, including the most famous sweeter one, Tokaji. Both restaurants have terraces, which may still be operating in October.
I wasn't there long enough to try more places, but some of the recommendations on this website looked like they would be worth the visit: