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We theoretically investigate a practically realizable Floquet topological superconductor model based on a one-dimensional Rashba nanowire and proximity-induced s-wave superconductivity in the presence of a Zeeman field. The driven system hosts regular 0-Majorana end modes and anomalous pi-Majorana end modes (MEMs). By tuning the chemical potential and the frequency of the drive, we illustrate the generation of multiple MEMs in our theoretical setup. We utilize the chiral symmetry operator to topologically characterize these MEMs via a dynamical winding number constructed out of the periodized evolution operator. Interestingly, the robustness of the 0- and pi-MEMs is established in the presence of on-site time-independent random disorder potential. We employ the twisted boundary condition to define the dynamical topological invariant for this translational-symmetry broken system. The interplay between the Floquet driving and the weak disorder can stabilize the MEMs, giving rise to a quantized value of the dynamical winding number for a finite range of drive parameters. This observation might be experimentally helpful in scrutinizing the topological nature of the Floquet MEMs. We showcase another driving protocol, namely, a periodic kick in the chemical potential, to study the generation of Floquet MEMs in our setup. Our work paves a realistic way to engineer multiple MEMs in a driven system.