We propose AdaDS, a generalizable framework for depth super-resolution that robustly recovers high-resolution depth maps from arbitrarily degraded low-resolution inputs. Unlike conventional approaches that directly regress depth values and often exhibit artifacts under severe or unknown degradation, AdaDS capitalizes on the contraction property of Gaussian smoothing: as noise accumulates in the forward process, distributional discrepancies between degraded inputs and their pristine high-quality counterparts diminish, ultimately converging to isotropic Gaussian prior. Leveraging this, AdaDS adaptively selects a starting timestep in the reverse diffusion trajectory based on estimated refinement uncertainty, and subsequently injects tailored noise to position the intermediate sample within the high-probability region of the target posterior distribution. This strategy ensures inherent robustness, enabling generative prior of a pre-trained diffusion model to dominate recovery even when upstream estimations are imperfect. Extensive experiments on real-world and synthetic benchmarks demonstrate AdaDS's superior zero-shot generalization and resilience to diverse degradation patterns compared to state-of-the-art methods.