The problem of super-resolution compressive sensing (SR-CS) is crucial for various wireless sensing and communication applications. Existing methods often suffer from limited resolution capabilities and sensitivity to hyper-parameters, hindering their ability to accurately recover sparse signals when the grid parameters do not lie precisely on a fixed grid and are close to each other. To overcome these limitations, this paper introduces a novel robust super-resolution compressive sensing algorithmic framework using a two-timescale alternating maximum a posteriori (MAP) approach. At the slow timescale, the proposed framework iterates between a sparse signal estimation module and a grid update module. In the sparse signal estimation module, a hyperbolic-tangent prior distribution based variational Bayesian inference (tanh-VBI) algorithm with a strong sparsity promotion capability is adopted to estimate the posterior probability of the sparse vector and accurately identify active grid components carrying primary energy under a dense grid. Subsequently, the grid update module utilizes the BFGS algorithm to refine these low-dimensional active grid components at a faster timescale to achieve super-resolution estimation of the grid parameters with a low computational cost. The proposed scheme is applied to the channel extrapolation problem, and simulation results demonstrate the superiority of the proposed scheme compared to baseline schemes.