Why did he ignore the old man?

The man ignored the old-timer, because the man is stupid and overconfident.  

Well, here he was; he had had the accident; he was alone; and he had saved himself. Those old-timers were rather womanish, some of them, he thought. All a man had to do was to keep his head, and he was all right. Any man who was a man could travel alone.

Ignoring the advice of an older, wiser, and more experienced person is a standard American motif.  It's why most of our heroes are young.  It's why most of our pop stars are young.  American's have always valued youth over experience.  Just think of movies like Ferris Bueller's Day Off. There isn't a single adult in that film that is portrayed in a positive light.  All of them are portrayed as dupes.  

That's the attitude that the man takes with the old-timer.  The man thinks that because he is younger and stronger that he will be fine.  He thinks that because he has traveled alone in other parts of the country that he will be fine this time. 

This fact did not worry the man. He was used to the lack of sun. It had been days since he had seen the sun, and he knew that a few more days must pass before that cheerful orb, due south, would just peep above the skyline and dip immediately from view.

The man is overconfident in his experiences, and believes that those experiences will transfer over to the Yukon environment without a problem.  The problem is that he is indeed new to the area and doesn't understand the importance of the little things that the environment is telling him. 

He was a newcomer in the land, aChechaquo, and this was his first winter. The trouble with him was that he was without imagination. He was quick and alert in the things of life, but only in the things, and not in their significances.

I believe another reason that the man ignores the old-timer is because the man is eager to join up with his mining buddies.  He knows that his colleagues are already on site, and he wants to get there quickly in order to help them mine.  

He was bound for the old claim on the left fork of Henderson Creek, where the boys were already.