Sampling-based anticipatory algorithms can be very effective at solving online optimization problems under uncertainty, but their computational cost may be prohibitive in some cases. Given an arbitrary anticipatory algorithm, we present three methods that allow to retain its solution quality at a fraction of the online computational cost, via a substantial degree of offline preparation. Our approaches are obtained by combining: 1) a simple technique to identify likely future outcomes based on past observations; 2) the (expensive) offline computation of a "contingency table"; and 3) an efficient solution-fixing heuristic. We ground our techniques on two case studies: an energy management system with uncertain renewable generation and load demand, and a traveling salesman problem with uncertain travel times. In both cases, our techniques achieve high solution quality, while substantially reducing the online computation time.