Accurate precipitation estimation is critical for flood forecasting, water resource management, and disaster preparedness. Satellite products provide global hourly coverage but contain systematic biases; ground-based gauges are accurate at point locations but too sparse for direct gridded correction. Existing methods fuse these sources by interpolating gauge observations onto the satellite grid, but treat each time step independently and therefore discard temporal structure in precipitation fields. We propose Neural Stochastic Process (NSP), a model that pairs a Neural Process encoder conditioning on arbitrary sets of gauge observations with a latent Neural SDE on a 2D spatial representation. NSP is trained under a single variational objective with simulation-free cost. We also introduce QPEBench, a benchmark of 43{,}756 hourly samples over the Contiguous United States (2021--2025) with four aligned data sources and six evaluation metrics. On QPEBench, NSP outperforms 13 baselines across all six metrics and surpasses JAXA's operational gauge-calibrated product. An additional experiment on Kyushu, Japan confirms generalization to a different region with independent data sources.