There is the Irish Centre on Digbeth High Street (B12 0LN) which is quite comfortable. A bit more on the way to the ground but a bit a more basic (well judging by the last time I visited) is the Waggon and Horses on Adderley Street (B9 4ED). As far as I am aware they both have Sky Sports.
I am just about to publish a page on the Football Ground Guide website about Coventry City at St Andrews. As a Birmingham City fan as well I have a pretty detailed knowledge of the area. Please take a look at it and let me know if anything needs changing or adding: Coventry City FC Groundshare...