We propose a data-driven and model-based approach to accelerate the Hamiltonian Monte Carlo (HMC) method in solving large-scale Bayesian inverse problems. The key idea is to exploit (model-based) and construct (data-based) the intrinsic approximate low-dimensional structure of the underlying problem which consists of two components -- a training component that computes a set of data-driven basis to achieve significant dimension reduction in the solution space, and a fast-solving component that computes the solution and its derivatives for a newly sampled elliptic PDE with the constructed data-driven basis. Hence, we achieve an effective data and model-based approach for the Bayesian inverse problem and overcome the typical computational bottleneck of HMC -- repeated evaluation of the Hamiltonian involving the solution (and its derivatives) modeled by a complex system, a multiscale elliptic PDE in our case. We present numerical examples to demonstrate the accuracy and efficiency of the proposed method.