Prices may go up from where they are now in downtown but there is also A LOT more parking which means competition. More people going downtown is a good thing, this adds vibrancy and energy which in turn brings people's money downtown and stimulates growth.
You say other business will get the shaft? More like they will benefit, restaurants, bars, hotels, etc will all benefit by having another 15,000-20,000 people near them for 150-200 nights a year. Why do you think businesses are clamouring for spots near the arena to build restaurants, bars, condos, etc? They know that people are going to spend time and money before and after games which benefits everyone and having your parking rate increase a little is a fair trade. One could argue that bringing in 4,000 more stalls will cause a drop in parking rates as it floods the market with more product (note that the City Centre mall is also adding 300+ stalls as well)
At any rate, there will be 13,500+ stalls within a 10 minute/800 m walk of the arena (4,000 being built by Katz), more than double that there is at Rexall and a lot will be covered/underground parking (much better and closer than the current parking).
http://icedistrictproperties.com/news/maps-graphics.html
People will also live and work near the arena negating the need to drive to the game/park as they will already be near the arena/can walk to the game. This is also true for out of town guest that are already staying downtown at hotels. There will also be more mass transit links to get to the game, especially once the Millwoods line is completed resulting in even less people needing to drive to the game.
I think parking on the east side of downtown near the Citadel, AGA, Winspear, etc won't be affected as much as they are on the further side (1 km away and outside the 13,500+ stalls mentioned above). Businesses and people will need to adjust at the beginning to see what works but I can't see everyone jacking their prices too much as there is simply too much supply and other means of transportation to get to the events.