Unified hyperspectral image (HSI) restoration aims to recover diverse degradations within a single model. However, current methods often rely on impractical explicit priors or opaque black-box representations that overfit to training distributions, hampering generalization to unseen scenarios. To bridge this gap, we propose Degradation-Aware Metric Prompting (DAMP), a novel framework that characterizes multi-dimensional degradations through interpretable spatial-spectral metrics. These metrics serve as Degradation Prompts (DP), enabling the model to capture shared characteristics across tasks and adapt to unknown corruptions. Central to our framework is the Degradation-Adaptive Mixture-of-Experts (DAMoE), where Spatial-Spectral Adaptive Modules (SSAMs) serve as experts that utilize learnable fusion coefficients to specialize in distinct degradation degrees. By using DP as a gating router, DAMoE dynamically activates specialized experts tailored to the specific degradation profile. Extensive experiments on natural and remote sensing HSI datasets demonstrate that DAMP achieves state-of-the-art performance and exhibits exceptional zero-shot generalization on unseen restoration tasks. Code is publicly available at \href{DAMP}{https://github.com/MiliLab/DAMP}.