Golden Ball was my not-so-local local when I lived in York and I watched many a match in there but that was some time ago now and it wasn't exactly set up for footie. Cracking pub though.
I usually head to Keystones these days. It used to be a student pub but now does craft beer and is best place to watch sport. If you're near the minster end of town I'd definitely recommend there. https://www.social-squirrel.com/thekeystonesyork
If you ask in Keystones they'll probably put it on one of the other TVs, they can show different games at the same time
They usually have 6 keg and 4 cask on. You can't go wrong for pubs in York! For a proper old style York pub try the Snickleway on Goodramgate. Duke of York in Kings Square is good and Pavement Vaults at the top of Piccadilly. Market Cat is a Thornbridge pub too. Brew York beer Hall is amazing on Walmgate and has over 40 beers on tap but unfortunately it's closed tomorrow!
I have very fond memories of time spent at that snug table in the window of The Snickleway, formerly The Anglers Arms. Great spot from which to watch people scurrying by, being busy.
It's been left behind a bit in choice of beers by other places but its still great. 10 minute walk from mine.