We study stochastic optimization with a completely unobserved target distribution and samples from related source distributions. Treating the sources and the target as independent draws from a common meta-distribution, we propose Wasserstein-metric meta-distribution support learning (MDSL). The model jointly learns the center and radius of a Wasserstein ball, with an exclusion parameter controlling the fraction of sources left outside. We identify a geometric failure of kernel-embedding-based descriptions, whose enclosing balls can admit arbitrarily dispersed distributions and produce infinite worst-case costs. The proposed MDSL is convex and admits a strong-duality formulation whose inner problem is a weighted Wasserstein barycenter of the sources. For a finitely supported learned center, standard Wasserstein duality yields a convex robust decision problem with separable inner maximizations. We connect these two stages through heuristics informed by MDSL: support distributions, their weights and optimal transport plans generate candidate scenarios for a finite convex initialization problem, while transport directions and available loss gradients guide search priorities. Cached candidates seed subsequent inner searches and are augmented by newly found adverse scenarios. On a compact observation space, we establish source-estimation perturbation bounds and, under quadratic growth, stability bounds for the center and admissible radius interval. We also derive out-of-distribution coverage and decision-performance guarantees for margin-enlarged balls, separating uncertainty from finitely many sources and finitely many observations per source. Under stated complexity and sampling conditions, coverage is asymptotically at least one minus the exclusion level as the enlargement vanishes. The framework thus links distributional support learning with robust optimization through both a learned uncertainty set and reusable transport information.