To me, your question sounds, as if you have trouble showing the gradual development of your character. A good discussion of how to provide well-rounded characters arcs is provided, for example, in
- Chris Vogler's The Writer's Journey: This was the most helpful book I've ever read about storytelling. It adresses the very essence of what a story is and how it works, i.e.: It discusses the dynamics of storytelling.
- K.M. Weiland's Structuring your Novel is much less concerned with the dynamics of a story and feels a bit formulaic at times. Still, it gives you a good idea of how to apply the infinite wisdom of the Hero's Journey (sorry for getting carried away with the Hero's Journey, I just love it). This book is more about practice and not so much about understanding.
Concerning your question (and speaking as someone who focuses very much on the development of characters), I would definitly be taken aback by an epiphany that comes out of the blue. Sometimes, this happens. However, I'm not interested in stuff that "just happens". I want to be presented with a thorough interpretation of why it happens.
The generic answer to this question is (and it really boils down to this in every single story): The transformation of the main character had to happen, because otherwise, he would have died. This is a drastic statement, but I truly believe it is true. Transformation is painful, tedious, and everything but comfortable. Think of the transformations you have gone through: Pubity sucks, losing weight (or gaining it) is a constant struggle, realizing that you will never be with the boy you are so helplessly in love with rips your heart apart. Nobody chooses to undergo a transformation because of all the fun it will be. People change, because they have no other choice.
That being said, you need to make sure that your main character has no other choice but to grow out of his fear. Build up the pressure he is under gradually and increase the strain his hesitation puts on him. Have him doubt his apathy. Then, when his entire world is on the brink of collapsing due to his passivity, make him realize it actually is in his power to prevent this dire fate. As far as I see it, this is the heart piece of your story. Hence, do not attenuate it by understressing it or not preparing it meticulously. In the words of Weiland: Make it big, make it fast.