Precise characterization of localized channel behaviors plays a vital role in the optimization of 5G cellular networks. However, many existing approaches either overlook fine-grained geographic variations or impose excessive computational burdens. To overcome these issues, this work introduces a localized statistical channel model (LSCM) that adapts to the propagation characteristics of a specific region. In contrast to CIR-based approaches, the LSCM framework reconstructs the channel using only reference signal received power (RSRP) measurements, expressed in a linear form with respect to the angular power spectrum (APS). This allows this channel representation to be considered as a sparse recovery problem, so nonzero APS components correspond to the power levels and departure directions of significant propagation paths. Furthermore, we introduce a joint block non-negative orthogonal matching pursuit (JBNOMP) algorithm that leverages block-structured sparsity and shared propagation characteristics across neighboring grids. Finally, simulation studies are carried out to demonstrate the performance advantages of the proposed approach.