I don't see any real show stoppers but if you fill it up with wine where do you keep your food!
Is there an AC vent in their already? A self contained standing AC unit might work if its a large enough pantry but they still have an exhaust hose that has to go outside or into a vent of sorts, otherwise your just defeating your purpose as you would be pouring heat into a room your trying to cool down. Inside rooms usually have less heat swings because they are away from the outer walls but quite often because they are inside a builder will cut corners and not put any insulation in the wall etc. If you can add any insulation anywhere in the room it would help to keep out the heat but it may not be pratical since its under a stairwell.
My winery/cellar is an 8x20 portion of the back part of my garage. Last Fall I added 12" of attic blanket up in the crawl space above and already I can tell its going to make a major difference this Summer in keeping that room cooler. I have a window as well as a floor heat vent to put some type of AC cooling in the room if I need it. I live in cool country but we usually have one month of hot weather in June. After that, we have monsoons that keep the rest of the Summer heat down most of the time.
Its the temp swings that really hurt the wine. Your wine will be much happier if you can manage to keep the room to stay within a 4-5 degree temp swing in a single day instead of a 10-15 degree. The wine will breath more with the larger temp swings thus aging it faster (even corked) And it goes without saying you should keep all your wine away and out of direct sunlight.
Pick up a portable electronic temperature gauge with a high/low memory from the hardware store and put it in there and see what your temp swings are. If you can keep the swings down it will do fine no matter what the temps are in the rest of the house.