I've thought for a while now that there aren't many places to watch football in the City Centre, without needing to book in advance and pay a deposit.
When people from outside Manchester ask me for recommendations for somewhere to watch, there is no one place that springs to mind where you can just walk in and watch a game. Remember Sports Bar on Quay Street? Shame that place closed - was a bit of a dump, but huge and never needed to book.
The only dedicated sports bars in town now are:
Directors Box, which I really like and has a good choice of beers but is very small so lucky to be able to book a table if there is a major event on.
The Box, on Deansgate, is huge - but you always need to book, or you'll just keep getting moved along when people who have booked the table you're sat at turn up. I recall having to move tables 3 times in the first half of the Arsenal v Bournemouth game last season, when I was just waiting for my mates to turn up.
And the Sports Bar in the Basement of Bierkeller in Printworks is decent, but can't remember if you have to book in there. Not been for a while.
To answer the OP's question - the nearest of those ^^ to NQ is going to be Bierkeller in the Printworks!