The interleaving of impermeable and permeable surfaces along a runoff flow path controls the hillslope hydrograph, the spatial pattern of infiltration, and the distribution of flow velocities in landscapes dominated by overland flow. Predictions of the relationship between the pattern of (im)permeable surfaces and hydrological outcomes tend to fall into two categories: (i) generalized metrics of landscape pattern, often referred to as connectivity metrics, and (ii) direct simulation of specific hillslopes. Unfortunately, the success of using connectivity metrics for prediction is mixed, while direct simulation approaches are computationally expensive and hard to generalize. Here we present a new approach for prediction based on emulation of a coupled Saint Venant equation-Richards equation model with random forest regression. The emulation model predicts infiltration and peak flow velocities for every location on a hillslope with an arbitrary spatial pattern of impermeable and permeable surfaces but fixed soil, slope, and storm properties. It provides excellent fidelity to the physically based model predictions and is generalizable to novel spatial patterns. The spatial pattern features that explain most of the hydrological variability are not stable across different soils, slopes, and storms, potentially explaining some of the difficulties associated with direct use of spatial metrics for predicting landscape function. Although the current emulator relies on strong assumptions, including smooth topography, binary permeability fields, and only a small collection of soils, slope, and storm scenarios, it offers a promising way forward for applications in dryland and urban settings and in supporting the development of potential connectivity indices.