It depends on the context of the situation. 'I am going to get my lunch', the possessive pronoun 'my' means that you already have ownership of the lunch and you're retrieving it from a certain location. In Japanese, it would be the same as saying 僕のお昼ご飯行ってこよう！If this was the intended purpose, then you are saying it correctly.
However, in a colloquial/casual sense, if you do not have lunch and you want to go purchase it, then you can eliminate the pronoun 'my' so the resulting sentence would be 'I am going to get lunch'. In a casual conversation, this usually indicates that you are leaving to purchase it.
Note that both ways are perfectly fine in a casual conversation, however; depending on the situation you should do well to remember which one to select.