Edit (some details added): Letting $U$ denote the complement of the set where any two coordinate planes intersect, $f$ is an isomorphism when restricted to $U.$ We therefore have that the restriction (i.e., pullback) of $f_{\ast}\mathbb{C}_Y[n-1]$ to $U$ coincides with $\mathbb{C}_U[n-1]$ (by proper base change if you like).
In order to conclude that $f_{\ast}\mathbb{C}_Y[n-1] \simeq IC_X,$ we now just need to check the support and cosupport conditions which uniquely define the intersection cohomology sheaf (together with the fact that its restriction to $U$ is the (shifted) constant sheaf). These conditions are similar to, but more restrictive than, the support and cosupport conditions for perverse sheaves.
I recommend looking at page 21 of the wonderful article by de Cataldo and Migliorini, which can be found at http://arxiv.org/abs/0712.0349 for a statement of these support and cosupport conditions (and figure 1 on page 25 for a visual illustration of the definition).
Since the fibers of $f$ consist of a finite number of points, the cohomology of the fibers is non-zero only in degree zero. This shows that the first condition (the support condition) is satisfied.
For the second condition (the cosupport condition), you can either derive it from the support condition using Verdier duality and the properness of $f,$ or you can simply note that an open ball in $\mathbb{C}^{n-1}$ has non-zero compactly supported cohomology only in degree $2n-2.$
