Actually, it is a difficult quite to fit into FORMAL grammar, but if you allow for the incompleteness (which is common in how people really speak), I don't see any rules broken.
The number of days is taking the place of an article. You could properly say THE day, or A day, or A FEW days, or A HUNDRED SIXTY days -- any of those has the same structure.
He says, "but I guarantee you that a hundred sixty days ridin' that way..." the a is part of the number. We frequently say "a hundred sixty" rather than "one hundred sixty" and accept either as correct, but the first way is less formal.
And the quote is a little more difficult because there is a connective that he does not say out loud and he doesn't finish his sentence. He means a hundred sixty days OF riding that way or a hundred sixty days THAT YOU WILL SPEND riding that way... A HUNDRED SIXTY modifies DAYS as an article would and DAYS acts as the subject of an incomplete statement -- we can assume that the speaker was beginning to say what 160 days riding would (or will) be like.
He says it that way because he's trying to put the young man's enthusiasm into proper perspective by emphasizing the difference between the effect of the one day they have apparently ridden so far in contrast to the effect of the 160 days of riding that are ahead (or that would be ahead). The numbers are very important to the meaning, so he has worded it to emphasize the difference in the numbers.