We introduce an adjoint-based reduced-order model framework for fast and accurate estimation of quantities of interest for many-query linear problems. The method builds a reduced-order model with respect to the adjoint problem, thus bypassing the solution of the primal problem and drastically reducing computational cost. It creates a surrogate model that is independent of the loading configurations. It enables fast evaluation across multiple load cases and the generation of virtual charts to support decision-making. Numerical experiments on the Poisson equation and a plane-stress elasticity problem demonstrate that the adjoint reduced-order model converges rapidly, outperforms its primal counterpart, and provides reliable estimates of the quantities of interest. Importantly, it is often more practical to parameterize a kernel function than an entire set of external loads, making the method generic and particularly suited for early-stage prototyping and design optimization.

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