Distributionally robust policy learning aims to find a policy that performs well under the worst-case distributional shift, and yet most existing methods for robust policy learning consider the worst-case joint distribution of the covariate and the outcome. The joint-modeling strategy can be unnecessarily conservative when we have more information on the source of distributional shifts. This paper studies a more nuanced problem -- robust policy learning under the concept drift, when only the conditional relationship between the outcome and the covariate changes. To this end, we first provide a doubly-robust estimator for evaluating the worst-case average reward of a given policy under a set of perturbed conditional distributions. We show that the policy value estimator enjoys asymptotic normality even if the nuisance parameters are estimated with a slower-than-root-$n$ rate. We then propose a learning algorithm that outputs the policy maximizing the estimated policy value within a given policy class $Π$, and show that the sub-optimality gap of the proposed algorithm is of the order $κ(Π)n^{-1/2}$, where $κ(Π)$ is the entropy integral of $Π$ under the Hamming distance and $n$ is the sample size. A matching lower bound is provided to show the optimality of the rate. The proposed methods are implemented and evaluated in numerical studies, demonstrating substantial improvement compared with existing benchmarks.