The mid point of the Raday utca restaurant row is a little over a km south east from the Buddha Bar Hotel; or about a 15 minute walk if you make no wrong turns. Good thing is, that if you do make a wrong turn you might see something great and unexpected.
One of my favorite places lies in the heart of another good restaurant zone, District VII. The place is Macesz Huszár which bills its self as a Jewish Bistro. The food is authentic Hungarian, the service is good and the neighborhood is worth the walk. In the same general areas you will also find: Vakvarju Pest, Café Vian (breakfast), Trófea Grill Étterem to name a few varied examples that I know are good depending on your mood. You will find the heart of this zone about 1km (15 minute walk) North East of your hotel.
Due north near the Basilica you can find places like Café Kor, Belvárosi Lugas Vendéglő, Rézkakas Bistro, Hungarikum Bistro: Again, plan on about a 1 km walk (15 minutes).
Much closer is the Vaci utca tourist row with lots of food, some very good, but none particularly cheap. Mostly tourist establishments which isn’t bad, after all you are a tourist.
And finally, fairly close to home, due east, is Múzeum Kávéház és Étterem which I believe serves breakfast (check me on that). It’s just 800m up the road.
As for the M3 Metro Stop nearby. It will get you very close to Raday utca , but it will only save you about 5 minutes and you will spend all that time underground where you cant anything. In the other direction its not much use to a tourist.
Fashion shopping? Andrassy ut or Deák Ferenc utca or Vaci utca. Don't miss il Bacio di Stile
Corrected in BOLD, thanks to Worldinbetween.