@Soccermumamir - right, here goes!
Very close to the Shambles and 5 mins from the Christmas market:
Duke of York on King's Square (top of shambles). Good food.
Pivni York on Patrick Pool - tiny place near the Shambles Market. Good continental beers and candlelit.
On Goodramgate, near the Brown's department store end of the Christmas market:
Royal Oak, Golden Slipper and the Old White Swan (locally known as the mucky duck) - very traditional and in ancient buildings. I like the Royal Oak best.
On Fossgate, round the corner from M&S at the bottom end of the Christmas market:
The Blue Bell - tiny, ancient place. Interior hasn't changed in decades (in a good way!). Worth a look if you can get in.
Also on fossgate is the Fossgate Tap and the Fossgate Social. The latter is more like a continental cafe bar and has a really nice courtyard out back.
Fossgate is worth a look anyway: pedestrianised, some nice places to eat and delis etc.
On Petergate very close to the minster:
Three Legged Mare
Eagle and Child
This is just off the top of my head - there are more!
York is very compact, so where something looks like a bit of a walk, it isn't really.
Hope this helps!