If you are told to use the contrapositive, you are NOT trying to reach a contradiction.
These two statements are equivalent: p ->q and ~q -> ~p. So if you can prove the second statement, then you prove the first one.
So since you are trying to show that if r^3 is irrational then r is irrational, one way to do this is to prove its contrapositive, that if r is rational, then r^3 is rational.
There is no disproof involved!