Generally... not as quickly. The salt content of the wine is much higher than drinking wine, which will preserve it longer. It will still "go bad" eventually. (Going bad is more a decrease in quality. It will still be edible, but unpleasant. It won't make you sick, except by the bad taste!)
See this from a Wikipedia Article: "Cooking wine typically available in North America is treated with salt as a preservative and food coloring."
So, it's safer than normal wine, but I'd try and use it soon. The sooner, the better.
Speaking from personal experience mulling wine many times:
In general, you want a dry or semidry red, of innocuous flavor profile. You do not want anything with strong tannic, acidic, alcoholic, brett or sweet flavors; these will become concentrated while mulling and quite unpleasant. Your ideal mulling wine is an inexpensive, inoffensive, young merlot, burgundy, petite syrah, tempranillo, beaujolais, or other "middle-of-the-road" wine, maybe slightly on the sweet side. Cabernet sauvignon, chianti, rioja, and similar wines tend to be poor choices, although of course it depends on the individual wine. Also, look out for high-sulfite-added wine which also can develop off flavors.
I'll contradict Sarge here and say that you do not want a wine which is turning towards vinegar; you'll end up with a very sour crock pot full of mulled vinegar. However, mulling is an excellent thing to do with wines which have been oxidized (but not vinegared) and lost a lot of their flavor, either through being open too long or too long on the shelf. Certainly if you spend more than $9 a bottle in the USA for wine for mulling, you've made a mistake.
This is very similar to how you would choose a sangria wine. The main difference is that for sangria you want bright and acidic flavors, whereas for mulled wine you want heavier, darker flavors.
Best Answer
The quickest way to get rid of leftover wine is to think of it as flavoured water. In many if not most recipes that call for water - especially stovetop recipes like sauces and stews - you can simply substitute wine for the water or stock that the recipe normally calls for.
We actually had a similar question recently: In what kind of recipes can I substitute stock for water? and I'd recommend you take a look at that, as many of the points there apply equally well to wine. Probably the best summation was in bikeboy389's answer:
Just swap the terms "stock" and "wine" and you're good to go. Even though wine doesn't contain any gelatin, you actually do need to be mindful of the gelatin aspect when substituting wine for stock, because you might have the opposite problem if the liquid is supposed to thicken.
Similarly to stock, I also wouldn't recommend using wine in anything that you plan to refrigerate or freeze for a long period, for a different reason obviously - because it can go sour over time. So try to only use it in recipes that will be consumed in the near future.
Other than that, just experiment; whenever you're making a savoury recipe that calls for water or stock, try using some wine instead. You'd be surprised at how much character it can add to otherwise simple dishes. Some examples of places where you can replace some other liquid with wine:
There really are no rules, and any time you find yourself tossing plain water into a pot or bowl, you should keep in mind that you are potentially missing an opportunity to add flavour (which a good wine will add plenty of).