The exact Riemann solver for one-dimensional elastic-perfectly plastic solid has been presented in the previous work [S. Gao and T. G. Liu, Adv. Appl. Math. Mech., 9(3), 2017, 621-650], but its iterative process of finding nonlinear equation solution is time-consuming. In this paper, to enhance the computational efficiency of the exact Riemann solver and provide a more practical Riemann solver for actual implementation, we design a non-iterative solution structure-based adaptive approximate (SSAA) Riemann solver for one-dimensional elastic-perfectly plastic solid. Judging the solution structure adaptively and then solving the Riemann problem with corresponding solution structure non-iteratively can shorten the computing time and meanwhile guarantee the correctness of the final result. Numerical performance tests manifest that the exact Riemann solver is indeed time-consuming and the ordinary approximate Riemann solver with fixed three-wave solution structure is of great error, whereas the SSAA Riemann solver is of both efficiency and accuracy. Error estimation further indicates that the SSAA Riemann solver has at least second-order accuracy to approach the exact solution of the states in the star region.