Machine learning has demonstrated transformative potential for database operations, such as query optimization and in-database data analytics. However, dynamic database environments, characterized by frequent updates and evolving data distributions, introduce concept drift, which leads to performance degradation for learned models and limits their practical applicability. Addressing this challenge requires efficient frameworks capable of adapting to shifting concepts while minimizing the overhead of retraining or fine-tuning. In this paper, we propose FLAIR, an online adaptation framework that introduces a new paradigm called \textit{in-context adaptation} for learned database operations. FLAIR leverages the inherent property of data systems, i.e., immediate availability of execution results for predictions, to enable dynamic context construction. By formalizing adaptation as $f:(\mathbf{x} \,| \,C_t) \to \mathbf{y}$, with $C_t$ representing a dynamic context memory, FLAIR delivers predictions aligned with the current concept, eliminating the need for runtime parameter optimization. To achieve this, FLAIR integrates two key modules: a Task Featurization Module for encoding task-specific features into standardized representations, and a Dynamic Decision Engine, pre-trained via Bayesian meta-training, to adapt seamlessly using contextual information at runtime. Extensive experiments across key database tasks demonstrate that FLAIR outperforms state-of-the-art baselines, achieving up to 5.2x faster adaptation and reducing error by 22.5% for cardinality estimation.