The widespread adoption of artificial intelligence (AI) in next-generation communication systems is challenged by the heterogeneity of traffic and network conditions, which call for the use of highly contextual, site-specific, data. A promising solution is to rely not only on real-world data, but also on synthetic pseudo-data generated by a network digital twin (NDT). However, the effectiveness of this approach hinges on the accuracy of the NDT, which can vary widely across different contexts. To address this problem, this paper introduces context-aware doubly-robust (CDR) learning, a novel semi-supervised scheme that adapts its reliance on the pseudo-data to the different levels of fidelity of the NDT across contexts. CDR is evaluated on the task of downlink beamforming where it outperforms previous state-of-the-art approaches, providing a 24% loss decrease when compared to doubly-robust (DR) semi-supervised learning in regimes with low labeled data availability.