There's nothing inherently special about the problems (and they're not even, say, math-contest-style problems); they're just intended to be reasonably comprehensive (with respect to an undergrad math education) and demonstrate the virtues of a written rather than oral examination. There's nothing there that anyone with a undergrad math degree should have any problem with, possibly modulo looking up a couple of specific formulas. He published a note after the problem set comparing it to other such examinations, if that's helpful to you.
That having been said, the subjects chosen aren't what I would expect. There's a lot of involved calculation, including some questions on numeric approximations. There are pages and pages and pages of questions about differential equations; in fact, the questions are heavily skewed to applied math in general. There are a couple of desultory questions about group theory and probability at the end, but that's about it; the rest is real and complex analysis (including a bunch of computations of specific integrals) and dozens of questions about differential equations. There's nothing about set theory, ring theory, commutative algebra, Galois theory, topology (beyond one question about Betti numbers and a couple about Riemann surfaces), Lie algebras (Problems #89 and #90 don't count), representation theory, any group theory that one wouldn't get in a physics class, etc. That would make more sense if Arnol'd was using it as a way to weed out prospective grad students working specifically under him, but it focuses on an extraordinarily narrow and applied curriculum. It doesn't strike me as a great way of demonstrating mathematical understanding.