According to the recent experimental progress of simulating energy band topology and dynamic quantum phase transitions (DQPTs) in ultracold atomic systems, we develop a periodically driven one-dimensional (1D) Raman lattice system to simulate dynamic topological phenomena. By utilizing amplitude-periodically modulated Raman beams to couple the $ \{^1{{\mathrm{S}}}_0, {}^3{{\mathrm{P}}}_1\} $ manifolds of alkaline-earth-like atoms $^{171}{\rm{Yb}}$ , we can realize the desired periodically driven Raman lattice. Utilizing the single-band, tight-binding Hamiltonian of the time- periodic system, we analytically determine the effective Floquet Hamiltonian and the micromotion operator. These allow us to investigate the conditions under which Floquet dynamic quantum phase transitions and dynamic skyrmion structures emerge at any driving frequency in the 1D Raman lattice. When the corresponding vector trajectory of the effective Floquet Hamiltonian has a non-zero winding number ( $\nu \neq 0$ ), the system exhibits both Floquet dynamic quantum phase transitions and dynamic skyrmion structures. For $\nu =0$ , Floquet dynamic quantum phase transitions may still occur, but dynamic skyrmion structures will definitely disappear. Therefore, the topologically nontrivial nature of the effective Floquet Hamiltonian is a sufficient but not necessary condition for the onset of the Floquet dynamic quantum phase transitions. But it is a necessary and sufficient condition for the onset of the dynamical skyrmion structures.