Learn Turkish: Yarın müdürü arıyorum. - I am calling the manager tomorrow.

Müdür means manager. We add the -ü ending because the manager is the specific object of the verb. If someone asked you Kimi arıyorsun? (Whom are you calling?), the answer requires this specific object ending (the accusative case), making it müdürü.

The verb root is ara- (from aramak). Why does it become arıyorum instead of arayorum?

When a verb root ends in -a or -e, that vowel narrows to ı, i, u, or ü when you add the -yor ending. Here, ara becomes arı to make the word flow better when spoken.

The word yarın means tomorrow, but arıyorum is present continuous (I am calling). Don't we need a future tense here?

You can use a future tense, but you don't have to! Just like in English (I am calling the manager tomorrow), Turkish frequently uses the present continuous tense (-yor) to talk about firmly planned actions in the future.

I learned that aramak means to search or look for. Does it also mean to call?

Yes! Aramak is used for both searching for something and calling someone on the phone. You can usually tell which one it is from the context.