I am supposed to determine if shocking either the income or preferences would certainly change the optimum solution in case of a corner solution?
I'm not sure what you mean by "shocking the preferences". But it's very easy to find an example in which the optimal solution is still at the corner after a (small) positive income shock.
Suppose $u(x,y)=\sqrt x+y$, $p_x=p_y=1$, and income is $m=0.1$. Performing utility maximization subject to budget constraint, we get $$ x^*=m,\qquad y^*=0. $$ Now give income a positive shock: $m'=m+\delta$. As long as $\delta\le0.15$, the optimal solution is still going to be a corner solution: spend all the income on $x$ and zero on $y$.