We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The interface is modeled by the graph of a function $φ: \mathbb Z^2 \to \mathbb Z$,and the disorder is given by a fixed realization of a field of IID centered random variables$(ω_x)_{x\in \mathbb Z^2}$. The Hamiltonian of the system depends on three parameters $α,β>0$ and $h\in \mathbb R$ which determine respectively the intensity of nearest neighbor interaction the amplitude of disorder and the mean value of the interaction with the substrate, and is given by the expression $$\mathcal H(φ):= β\sum_{x\sim y} |φ(x)-φ(y)|- \sum_{x} (αω_x+h){\bf 1}_{\{φ(x)=0\}}.$$ We focus on the large-$β$/rigid phase phase of the Solid-On-Solid (SOS) model. In that regime, we provide a sharp description of the phase transition in $h$ from a localized phase to a delocalized one corresponding respectivelly to a positive and vanishing fraction of points with $φ(x)=0$. We prove that the critical value for $h$ corresponds to that of the annealed model and is given by $h_c(α)= -\log \mathbb E[e^{αω}]$, and that near the critical point, the free energy displays the following critical behavior $$F_β(α,h_c+u )\stackrel{u\to 0+}{\sim} \max_{n\ge 1} \left\{θ_1 e^{-4βn} u- \frac{1}{2}θ^2_1 e^{-8βn} \frac{\mathrm{Var}\left[e^{αω}\right]}{\mathbb E \left[ e^{αω} \right]^2}\right\}.$$ The positive constant $θ_1(β)>0$ is defined by the asymptotic probability of spikes for the infinite volume SOS with $0$ boundary condition $θ_1(β):=\lim_{n\to \infty} e^{4βn}\mathbf P_β (φ({\bf 0})=n)$ ...