Large language model (LLM)-based conversational recommender systems (CRSs) have demonstrated strong capabilities in capturing user preferences and generating contextually relevant recommendations. Nevertheless, the recommendation quality of the models frozen after training inevitably degrades under contextual shifts, such as changes in language and social trends. While periodic model updates are essential to maintain alignment with real-world preferences, training on large-scale data incurs substantial costs. This motivates data-efficient adaptation. However, existing data selection methods struggle to distinguish learnable samples under contextual shifts. To address this, we propose Contextual Shift-Adaptive Data Pruning and Training (CAPT), a framework agnostic to underlying LLM-based CRSs. Specifically, we conceptualize a three-class data taxonomy comprising familiar, valuable, and outlier samples to formalize data behavior under contextual shifts. Based on this taxonomy, we design an importance score estimation scheme that quantifies a sample’s relative learnability for shift adaptation. Leveraging these importance scores, CAPT prioritizes highly learnable samples and further guides shift-adaptive training to actively steer the model toward evolving preferences. Experiments on three CRS benchmarks with real-world temporal splits demonstrate that CAPT outperforms baselines, matching or surpassing full-data fine-tuning performance using only 10-50% of the training data.