The answer to this question is quite simple: Julie/Miyax overcomes fear by concentrating on her survival. Allowing fear to overcome her means death to Julie. She must keep her wits about her because the Arctic winter is coming. Julie originally planned to escape to San Francisco in order to find her pen pal, Amy. Now that she has lost her way, she reverts to the old Eskimo ways that she knows so well (from living at the seal camp with her father). In order to take away her fear, Julie continually remembers her father's advice. Kapugen's advice is especially useful when she comes upon the wolf pack:
Wolves are brotherly. ... They love each other, and if you learn to speak to them, they will love you too.
More generally, we can see Julie overcome fear many times in the book. Julie builds a sod hut as soon as she realizes she is lost. Julie is very patient as she learns the wolves' mannerisms. Julie must have confidence in herself in order to become part of the pack. Once Julie realizes that lying belly-up is the ultimate submission (and what the pups always do to Amaroq), Julie uses this technique to submit to Amaroq. This bravery on Julie's part is what makes her part of the pack.
As you can see, in all of the instances above, Julie overcomes her fear by reverting to the "old ways" of the Eskimo and not allowing her mind to wander back to the dangerous feeling of fear.