Learn Russian: Возьми зонт на всякий случай.

В случае means "in case of" and must be followed by a specific condition, like a noun in the Genitive (в случае дождя — in case of rain) or a clause (в случае, если...). Because this sentence doesn't state the specific condition, it uses the fixed standalone idiom на всякий случай (literally "onto every case"), which translates to "just in case".

Can I translate the English literally and just say Возьми зонт в случае?

No, that sounds incomplete in Russian. While English allows "in case" to stand alone at the end of a sentence, Russian requires в случае to be followed by its condition. If you don't want to specify the condition, you must use the complete idiom на всякий случай.

What case is на всякий случай in, and why?

It is in the Accusative case. The preposition на followed by the Accusative is often used to show preparation, provision, or purpose for a future event. Notice that this contrasts with the lesson's phrase в случае, which uses the Prepositional case.

Why is the perfective imperative возьми used here instead of the imperfective бери?

The speaker is requesting a single, complete, concrete action to be done right now: grabbing the umbrella to achieve the result of having it with you. The imperfective бери would sound like general advice ("always take an umbrella") or an impatient urging command if the person were hesitating.