The traditional frequency-modulated continuous wave (FMCW) multiple-input multiple-output (MIMO) radar two-dimensional (2D) super-resolution (SR) estimation algorithm for target localization has high computational complexity, which runs counter to the increasing demand for real-time radar imaging. In this paper, a fast joint direction-of-arrival (DOA) and range estimation framework for target localization is proposed; it utilizes a very deep super-resolution (VDSR) neural network (NN) framework to accelerate the imaging process while ensuring estimation accuracy. Firstly, we propose a fast low-resolution imaging algorithm based on the Nystrom method. The approximate signal subspace matrix is obtained from partial data, and low-resolution imaging is performed on a low-density grid. Then, the bicubic interpolation algorithm is used to expand the low-resolution image to the desired dimensions. Next, the deep SR network is used to obtain the high-resolution image, and the final joint DOA and range estimation is achieved based on the reconstructed image. Simulations and experiments were carried out to validate the computational efficiency and effectiveness of the proposed framework.