Is it safe to store applejuice in an opened glass bottle outside of the fridge? I've got a few glass bottles of clear applejuice in the cellar (just pure clear store brand applejuice) that I want to take to my computer room so I don't need to constantly go downstairs to refill my glass.

can I just put them on my desk and take refills from them over a 2-3 day period with the bottle being at room temperature and not stored in the shade? or would that detract from the quality, taste or healthiness of the drink?
Most of the apple juice sold in the US at least is Pasteurized. So unopened, you should have no problem storing at room temp. Once you open the bottle, microorganism's can get in. With a sugary liquid like Apple juice, wild yeasts will find a happy home, and ferment up a bubbly alcoholic apple beverage for you. This can take a week or two to happen, and the result is usually tasty rather than deadly. Sometimes you'll end up with vinegar rather than an alcoholic ferment.
There's some small chance of picking up a nasty microbe in your apple juice over a three day period, but it was never enough to stop people from drinking unrefrigerated cider in the days before Pasteurization or refrigeration.