Learn Turkish: Pasaport eksik, onu arıyorum. - The passport is missing, I am looking for it.

The verb aramak (to look for) takes a direct object. Because we are looking for a specific thing (the passport), we have to use the accusative case. The pronoun o (it) gets a buffer letter n and the accusative suffix u, making it onu.

I thought aramak meant "to call" on the phone. Can it mean "to look for" too?

Yes, aramak is used for both! You can use it to say you are calling someone or searching for something. Here, because passports don't have phones, the context tells us it means "looking for".

Why doesn't Pasaport have a suffix if the English translation says "The passport"?

Turkish doesn't have a direct translation for the word "the". When a word is the subject of the sentence (the thing being described as missing), it stays in its plain dictionary form. You only add suffixes for "the" when the word is a specific direct object.

Where did the "a" at the end of the verb root ara- go in arıyorum?

This is an example of vowel narrowing. The verb root is ara-. When we add the present continuous suffix -ıyor, the ending a of the root narrows into an ı to connect smoothly. So ara- plus -ıyor plus -um (I) becomes arıyorum.