I think we're forgetting one very important fact here. This fight is taking place in icicle inn. This match doesn't go to whoever can fight, it goes to whoever can fight in the snow.
Shinra Manager is a Midgar man through and though. Yes, he does go to Junon, but let's be honest, even that is something of an ordeal for him. He likes comfort, he likes respect, he likes people to follow the rules. Put him somewhere where people throw snowballs and go sledging for fun, somewhere where Shinra presence barely touches, and he'll be so far out of his comfort zone that nothing else will matter. Oh, he'll bluster 'you can't touch me, I'm a Shinra Manager!' and the like, but he's got no bite behind his bark.
Meanwhile the Deputy Mayor looks far more well travelled. He's older, but he's not old. In fact, he's in that sweet spot where vigour and strength is undiminished, but he has a wealth of experience to draw upon.
And as Fancy astutely observed, he sports the clothes of a martial artist, suggesting he's had at least some training. He's less rigid about following 'the rules'. He's not afraid to ask Avalanche for money, something that shows guts considering they're known terrorists sporting giant-ass swords and guns. He spends most of his time in a library filled with books about science, weaponry, urban planning and space travel. We can assume he's pretty well read, and will have no difficulty adapting to the situation he finds himself.
Plus he bears to loyalty to Shina, indeed he even carries a certain amount of resentment. He'll leap upon this opportunity to express all that frustration and long-standing bitterness -- he'll be on Shinra Manager before the Manager even works out what's going on.