"I get annoyed because of him!"
Translation:그 때문에 짜증이 나!
It really threw me that the final verb was just 나, since there is an unspoken 'I' in the translation as well. It might be better to use 해요체 to avoid confusion here
짜증나 is by far the most common way to hear this, to be fair. More 나다/내다 verbs set up together would be good for the course in fact, since they have an important distinction.