Real-world scientific applications frequently encounter incomplete observational data due to sensor limitations, geographic constraints, or measurement costs. Although neural operators significantly advanced PDE solving in terms of computational efficiency and accuracy, their underlying assumption of fully-observed spatial inputs severely restricts applicability in real-world application. We introduce the first systematic framework for learning neural operators from partial observation. We identify and formalize two fundamental obstacles: (i) the supervision gap in unobserved regions that prevents effective learning of physical correlations, and (ii) the dynamic spatial mismatch between incomplete inputs and complete solution fields. Specifically, our proposed LANO (Latent Autoregressive Neural Operator) introduces two novel components designed explicitly to address the core difficulties of partial observations: (i) a mask-to-predict training strategy that creates artificial supervision by strategically masking observed regions, and (ii) a Physics-Aware Latent Propagator that reconstructs solutions through boundary-first autoregressive generation in latent space. Additionally, we develop POBench-PDE, a dedicated and comprehensive benchmark designed specifically for evaluating neural operators under partial observation conditions across three PDE-governed tasks. LANO achieves state-of-the-art performance with 18--69$\%$ relative L2 error reduction across all benchmarks under patch-wise missingness with less than 50$\%$ missing rate, including real-world climate prediction. Our approach effectively addresses practical scenarios involving up to 75$\%$ missing rate, to some extent bridging the existing gap between idealized research settings and the complexities of real-world scientific computing.