We consider the problem of high-dimensional channel estimation in fast time-varying millimeter-wave MIMO systems with a hybrid architecture. By exploiting the low-rank and sparsity properties of the channel matrix, we propose a two-phase compressed sensing framework consisting of observation matrix completion and channel matrix sparse recovery, respectively. First, we formulate the observation matrix completion problem as a low-rank matrix completion (LRMC) problem and develop a robust rank-one matrix completion (R1MC) algorithm that enables the matrix and its rank to iteratively update. This approach achieves high-precision completion of the observation matrix and explicit rank estimation without prior knowledge. Second, we devise a rank-aware batch orthogonal matching pursuit (OMP) method for achieving low-latency sparse channel recovery. To handle abrupt rank changes caused by user mobility, we establish a discrete-time autoregressive (AR) model that leverages the temporal rank correlation between continuous-time instances to obtain a complete observation matrix capable of perceiving rank changes for more accurate channel estimates. Simulation results confirm the effectiveness of the proposed channel estimation frame and demonstrate that our algorithms achieve state-of-the-art performance in low-rank matrix recovery with theoretical guarantees.