This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each other are known, and target nodes, for which complete but corrupted distance measurements to the anchors are available. To tackle this problem, we propose a novel algorithm powered by Nyström method and robust principal component analysis. Our method is computationally efficient as it processes only a localized subset of the distance matrix and does not require distance measurements between target nodes. Empirical evaluations on synthetic datasets, designed to mimic sensor localization, and on molecular experiments, demonstrate that our algorithm achieves accurate recovery with a modest number of anchors, even in the presence of high levels of sparse outliers.