Is Korean cucumber soup a main dish or a side dish?

Oi naengguk is a side dish. In Korean meals, it comes to the table alongside rice and other side dishes. You'll often see it served at Korean BBQ restaurants too.

Can I make this cold cucumber soup ahead of time?

You can make the broth ahead of time. Just add the cucumber and ice right before serving, that way it tastes freshly made. But honestly, if you're eating it that same day, it's totally fine to make it a few hours ahead too.