That's right, this sentence means "he" will overcome the current trouble or difficult situation. Let's imagine the difficult situation as a high mountain in his way, so when he "gets over" the "mountain", it means he gets past the difficult situation. I think it is a more vivid picture and may help you understand this expression "get over something" more easily.

get over = to recover from

thank you.

The subject 'he' is having some kind of trouble (mostly emotional) but sometimes physical too. The phrase "He'll get over it" is to describe that he will overcome the situation.