I dont know how adventerous you are and what your plans are for that day. But if you take the beautiful walk up Andrassy ut in the evening (which is a great idea) then after you see the sights and you start heading back down Andrassy ut by foot or metro there are a few options that are very good.
https://kiraly100.hu/en/king100restaurant/ well over 100 years old and 3 owners in that time. Good Hungarian food, exceptional atmosphere. Very authentic.
https://www.andrassygarden.com/ another very traditioal menu done is a pretty special setting (good weather place)
https://paprikavendeglo.hu/ Ive been going here, I bet 20 years. Doesnt much more Hungarian. A bit of a walk off Andrassy ut, but you can stop at the House of Music and make the walk worth it.
https://www.ketszerecsen.hu/ another Hungarian tradition. More locals than tourists, but popular with the tourists and very close to my favorite wine bar