The old-timer from Sulphur Creek had warned the man of the dangers of going into the frigid Yukon Territory in the winter. He had tried to emphasize "how cold it sometimes got in the country", and outlined some basic survival skills the man would have to know in order to survive under those extreme conditions. The man "had laughed at (the old-timer) at the time", and, thinking he could handle it, had set off into the Alaskan wilderness in the winter alone.
Two especially critical pieces of advice the old-timer had given the man were the absolute necessity of being able to build a fire immediately should his feet get wet, and the wisdom of taking with him a companion. About the fire, the old-timer had warned, "...there must be no failure. When it is seventy-five below zero, a man must not fail in his first attempt to build a fire - that is, if his feet are wet...the circulation of wet and freezing feet cannot be restored by running when it is seventy-five below". The man remembers and appreciates the advice the old-timer gave him when he is in trouble, but had not taken it seriously enough to make sure he thoroughly learned how to make a fire under dire conditions before he set out.
When he cannot make a viable fire, the man regrets not having listened to the other element of the old-timer's advice - "If he had only had a trail-mate he would have been in no danger. The trail-mate could have build the fire".
You have hit the nail on the head when you say that it is the "old-timer from Sulphur Creek" who gave the man "key advice." It is so very "key" that, if the man would have listened, he would have still had his life at the end of the story instead of freezing in the frigid temperatures.
It is simply luck that the man meets the old-timer before his adventure in the Yukon. In the back of his mind, the man knew the Yukon in the winter was dangerous. The old-timer continually emphasized the cold, especially in "the country." The man is already in trouble in that he "laughed at him at the time." Still, the old-timer goes on with the most important part of his "key advice" you speak of in your question. He mentions the importance of taking someone else along (so that the man isn't in the wilderness alone). He mentions the importance of keeping one's feet dry. Then he says this about the cold:
There must be no failure. When it is seventy-five below zero, a man must not fail in his first attempt to build a fire - that is, if his feet are wet...the circulation of wet and freezing feet cannot be restored by running when it is seventy-five below.
We should get from this how absolutely imperative it is to build a fire right away in case one's feet get wet. Again, the man laughs. Full of pride in his own abilities, and unwilling to listen to the wisdom of the elderly, the man sets off.
Unfortunately for the man, he only takes into account the wisdom of the old-timer when he is approaching death, freezing in the extreme cold of the Alaskan wilderness. How do we know, as readers, that the man finally realizes the wisdom of the old timer? Because the man, dying, says this:
If he had only had a trail-mate he would have been in no danger. The trail-mate could have build the fire.
Thus, the man dies alone in the wilderness due to his own negligence.
The old timer from Sulfur Creek warned him not to travel outside alone when the temperature is below a certain point. The man's arrogance takes over and he feels the advice is "womanish". His pride goes to far as he bets his survival on his ability to survive the vicious cold. His mistake was first in believing himself superior to power of a cold and indifferent universe. It is only when he faces certain death that he relents. He then changes his perception of the old timer.
The old timer’s advice is to never travel alone during a cold snap. Specifically, London writes that the man “remembered the advice of the old man on Sulphur Creek, and smiled. The man had been very serious when he said that no man should travel alone in that country after 50 below zero.” The old timer specifically warned about the danger of getting one’s feet wet in such extreme cold, and about how “in the country” temperatures can be much more extreme. The man’s reaction to this advice is telling. In fact, the story’s unique, dispassionate point of view leaves out more than it tells. We get only the most basic reactions from the man: when he gets a fire going, he thinks the old man “womanish”; when the snow extinguishes his fire, he thinks that “perhaps” the old man had been right. The man begins with the confidence that he can ignore the old man’s advice; what we learn as the story progresses is how tragically wrong he is, both about his own endurance and his problem-solving skills.
An interesting exercise would be to retell the story from the old man’s point of view; for him, the man must have been just another in a long line of “greenhorns” who will either learn to adapt to the harsh climate of the Yukon or die out of hubris.