In this paper, we consider near-field localization and sensing with an extremely large aperture array under partial blockage of array antennas, where spherical wavefront and spatial non-stationarity are accounted for. We propose an Ising model to characterize the clustered sparsity feature of the blockage pattern, develop an algorithm based on alternating optimization for joint channel parameter estimation and visibility region detection, and further estimate the locations of the user and environmental scatterers. The simulation results confirm the effectiveness of the proposed algorithm compared to conventional methods.