Why does day old rice make for a better tasting fried rice?
The goal is to sufficiently dry it, so that you get the right texture of your fried rice. You want it to be noticeably dry, so that it will fry rather than steaming, and not clump together. It's okay if it's clumped up as it dried, as long as you can break it up without it sticking back together.

It does not actually have to be stale (changes in the starch that happen over time), just dry. So it's not exposure to oxygen, it's just moisture evaporating, i.e. exposure to air that's going to happen no matter how it's stored.

There are a lot of ways to accomplish this, including refrigerating for a day (or even 12 hours). There's no need to leave it uncovered. That might let you get away with a shorter time, so you could try it as a way to rush, but be aware that it may pick up smells from your fridge.

You can also use a fan to dry it quickly at room temperature. Simply leaving it at room temperature for a long period is unsafe, though, so I wouldn't recommend that.

See this recent answer and the Serious Eats article on fried rice it's largely based on for more details and related information.