Learn Hardcore Hungarian: Mindjárt felhívom a barátnőmet, csak előbb befejezem az e-mailt. - I will call my girlfriend in a moment, I just need to finish the email first.

In this sentence, it signals that the phone call will happen almost immediately, but not necessarily this exact second. It is often a little softer than azonnal, which is more like immediately.

So here mindjárt fits the idea of: I’m about to do it, just let me finish one thing first.

Why are felhívom and befejezem in the present tense if the meaning is future?

Hungarian very often uses the present tense to talk about the future when the time is already clear from context.

Here, words like mindjárt and előbb make the timing obvious, so Hungarian does not need a separate future form.

felhívom = literally I call it/her, but here it means I’ll call
befejezem = literally I finish it, but here it means I’ll finish

This is completely normal Hungarian. English usually prefers will, but Hungarian often does not.

Why is felhívom one word, and what does fel- do?

Felhívom is a verb with a prefix:

hív = to call, invite