Language understanding in the brain is context-dependent, varying across experimental stimuli and individuals, which makes it difficult to build computational models that generalize across both. This calls for a foundation model of language-evoked brain activity that can capture shared structure while adapting efficiently to new participants and inputs. We introduce RABBiT (Rapidly Adaptive BOLD foundation model via BraIn-Tuning), a compact audio-to-fMRI encoder designed for accurate zero- and few-shot prediction. A comprehensive evaluation on 324 participants across multiple unseen fMRI datasets shows that RABBiT enables accurate zero-shot prediction of fMRI responses to natural speech across auditory and language-selective regions, surpassing the SOTA foundation model for fMRI and predictions based on group averages. With as little as 10 minutes of participant-specific data, RABBiT further improves performance via parameter-efficient tuning, substantially outperforming per-participant linear models. RABBiT's performance is driven by two key innovations: (1) learned region-specific attention, and (2) a decomposition of brain responses into shared and subject-specific components, combined with a brain-tuned speech backbone. In addition to supporting strong predictive accuracy, the structured, region-specific representations that RABBiT learns enable interpretability. By eliminating the need for extensive per-participant data and model fitting, RABBiT enables scalable population-level analyses of language in the human brain. We make the code available at https://github.com/bridge-ai-neuro/rabbit.