Visit The House of Trembling Madness. The bar is above the shop on Stonegate.
You have a really good choice of placed to eat, but I’d recommend that you have a look at the places on Fossgate/Walmgate. Barbikan (Polish), Walmgate Ale House (English), The Hop (Wood fired Pizza), Khao San Road (Thai), Loch Fyne (Fish) and Mumbai Lounge (Indian) to name just a few. The Fossgate social is also very nice for a coffee.
Must also recommend Rustique (French), but usually have to book ahead.
So many pubs.
Just have a wander and see what you find. York is relatively compact, so quick to return to somewhere later on if you wish.
Enjoy