Learn French: Si c'est le cas, appelle-moi. - If that is the case, call me.

Good catch! While the standard first conditional uses si + present -> future, you can also use a command (the imperative) for the result. Here, appelle-moi (call me) is a command. The pattern is simply: if the condition is met, give this order!

Why is the pronoun at the end in appelle-moi instead of before the verb?

When you give a positive command (telling someone to do something), object pronouns attach to the end of the verb with a hyphen. Note that we use the stressed pronoun moi instead of me here. If it were a negative command, the pronoun would go back in front: ne m'appelle pas (don't call me).

If I am telling "you" (tu) to call me, why doesn't appelle end with an "s"?

For -er verbs like appeler, the tu command form drops the final "s". So instead of the standard present tense tu appelles, the command is just appelle.

Is le cas used the exact same way as "the case" in English?

Yes! Si c'est le cas translates perfectly to "If that is the case" or "If so". It is a very common, natural phrase to use in French when referring back to a situation someone just mentioned.