In the aftermath of a disaster, whether natural, industrial, or war-related, a rapid and accurate assessment of building damage is crucial for rescue forces to conduct an effective emergency response. Very high-resolution satellite imagery enables such assessments and serves as an important indicator for understanding the scale of destruction, supporting time-critical rescue operations, and guiding resource allocation. While deep learning models have shown promising results in automating building damage assessment (BDA) from pre- and post-disaster optical satellite imagery, they often fail to generalize to new disasters due to domain shifts. This paper studies the challenge of rapid domain adaptation for BDA in the context of the war in Ukraine. We create a new, challenging dataset annotated with damage grades across six cities in Ukraine, using pre- and post-disaster optical imagery. To facilitate rapid adaptation, we propose an efficient fine-tuning workflow using Low-Rank Adaptation. Our experiments show that this approach substantially improves performance in both out-of-domain and in-domain settings, presenting a practical and data-efficient study for deploying BDA models in time-critical emergency scenarios.