The Gordon Hotel/Arms on Wellington Road will be in walking distance of the stadium, not a 5 minute stroll obviously, but comparable with the distance between Archie Simpsons and Pittodrie. I'd imagine there'd be a bar in the Thistle Hotel too, albeit a bit pricey. Not familiar with Kincorth but I'd presume if a stadium is being built in their back garden, there'd be plenty of access built for local residents opening up access to their pubs. If the Cove bars have any sense, they'd lay on shuttle buses to get the punters in.
Building a pub onsite may not be a money spinner as mentioned above, but if they made it a Beefeater-type restaurant with a bar, that could be different. Mill Of Mundurno and the one across the A90 from Portlethen do well despite them being inaccessible to pedestrians.
Obviously none of those suggestions are ideal and as convenient for drinking as Pittodrie currently is, but there's plenty of opportunities to 'make do' for those who can't watch a game of football without getting blootered.