Time integration with stress–strain updating is a key step for the application of elasto-viscoplastic models to engineering practice. Currently, the estimation robustness of algorithms is lacking, which poses difficulties in the selection and improvement of algorithms. To solve this, the study selected four typical implicit time integration algorithms (i.e., Newton-Raphson, Katona, Stolle, and cutting plane) for the same simple elasto-viscoplastic modified Cam-clay model (EVP-MCC). Some necessary enhancements are discussed that were made for the integration. A series of laboratory tests was simulated, based on which the variations of the relative errors of stresses and iteration numbers with step size were investigated and compared. For the Newton-Raphson algorithm and the Katona algorithm with $\theta =0.5,\hspace{0.17em}1.0$ , the maximum step sizes ensuring convergence were found to be at least one order of magnitude larger than those of the other algorithms, and their total iteration numbers and relative errors of stresses were at least one order of magnitude lower than those of the other algorithms. Furthermore, the model using different algorithms was implemented in a finite-element code, and the global convergence and calculation time were investigated for a boundary-value problem. The robustness of all algorithms was estimated based on the calculation performance in terms of convergence, accuracy, and efficiency. The results demonstrate that the global iteration number for the cutting-plane algorithm is at least 20 times higher than the others at any mesh density, which leads to the result that the central processing unit (CPU) time for the cutting-plane algorithm is almost 10 times higher than the others. All comparisons demonstrate the performance of different time integration algorithms with a prior order of Newton-Raphson, Katona, Stolle, and cutting-plane algorithms.