Okay, I think I've got it figured out, and I think that Porteiro was right—either "on" can be dropped.
It's a cleft sentence, meaning it follows the form "it is x that/which y," as in "it is dogs that chase cats." You can turn it into a regular sentence by simply casting it as "x y"—"dogs chase cats." You can also switch around the order, like so: "it is cats that dogs chase."
The plain form here would be "our success in life depends on that." I'm going to replace "that" with the noun phrase it stands in for, "the same test," to reduce ambiguity. So now we have "our success in life depends on the same test." If you switch around the order first, you could have either "on the same test our success in life depends" or "the same test our success in life depends on."
Then transform it into a cleft sentence, and you get either "it is on the same test that our success in life depends" or "it is the same test that our success in life depends on." If you don't want to strand the preposition, you've got two choices of where to put it: before "the same test" (which is what option 1 has already done) or before "that," though it then becomes "which." That would leave us with "it is the same test on which our life depends." You can put the preposition with the "that/which" because it's a relative pronoun standing in for the x component, which is why it can go with either.
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Well, phooey. I thought I had it worked out, but then Ruth went and confused me again. She made the valid point that the preposition "on" has to go with the subordinate clause beginning with the relative pronoun "that/which." If you leave it out, you get the ungrammatical *that our success in life depends. And yet I still get the feeling that "for it is on that which our success in life depends" is perfectly grammatical. Maybe I'll e-mail someone who knows more about syntax than I do.
Also, I don't care if nobody here understood anything past the first sentence. I just had to try to work it out for my own sake.