While mirror symmetry for flag varieties and Grassmannians has been extensively studied, Schubert varieties in the Grassmannian are singular, and hence standard mirror symmetry statements are not well-defined. Nevertheless, in this article we introduce a ``superpotential'' $W^λ$ for each Grassmannian Schubert variety $X_λ$, generalizing the Marsh-Rietsch superpotential for Grassmannians, and we show that $W^λ$ governs many toric degenerations of $X_λ$. We also generalize the ``polytopal mirror theorem'' for Grassmannians from our previous work: namely, for any cluster seed $G$ for $X_λ$, we construct a corresponding Newton-Okounkov convex body $Δ_G^λ$, and show that it coincides with the superpotential polytope $Γ_G^λ$, that is, it is cut out by the inequalities obtained by tropicalizing an associated Laurent expansion of $W^λ$. This gives us a toric degeneration of the Schubert variety $X_λ$ to the (singular) toric variety $Y(\mathcal{N}_λ)$ of the Newton-Okounkov body. Finally, for a particular cluster seed $G=G^λ_{\mathrm{rec}}$ we show that the toric variety $Y(\mathcal{N}_λ)$ has a small toric desingularisation, and we describe an intermediate partial desingularisation $Y(\mathcal{F}_λ)$ that is Gorenstein Fano. Many of our results extend to more general varieties in the Grassmannian.