Ranking nodes in a multiplex network is one of the most pressing and challenging tasks in network analysis. Generalizing centrality measures to multiplex networks is an active area of research. The Multiplex PageRank model is an extension of Google's PageRank, which introduces a new centrality measure to extend the usual PageRank to multiplex networks. In this work, we focus on the Multiplex PageRank problem. First, based on the special structure of the Multiplex PageRank problem, we propose an inverse-free block-SOR method. Second, with the help of randomly sampling, we propose a new strategy for estimating the optimal relaxation parameter. Specifically, the multiplex network is frequently updated in real world applications, and we have to deal with temporal Multiplex PageRank problems including the incremental and the decremental Multiplex PageRank problems. To the best of our knowledge, however, there are few efficient algorithms for solving these type of problems. To fill-in this gap, the third contribution of this work is to propose both incremental and decremental algorithms for solving the temporal Multiplex PageRank problems. Comprehensive numerical experiments are performed to illustrate the numerical behavior of the proposed algorithms, and show the effectiveness of our new strategies.