A slant ruled surface is a unique type of ruled surface composed by Frenet vectors that form a constant angle with each other and with specific directions in space. In this paper, the non-null slant ruled surface, which is generated by the striction curve of the natural lift curve, was constructed with a novel approximation in $E_1^3$. To establish the approximation, E. Study mapping was then applied to determine the relationship between pseudo-spheres and non-null slant ruled surfaces that are generated by the striction curves of the natural lift curves. Furthermore, $\overrightarrow{\overline{q}}-,\overrightarrow{\overline{h}}-,\overrightarrow{\overline{a}}-$ spacelike (resp., timelike) slant ruled surfaces were classified by using the striction curves of the natural lift curves in $E_1^3$. We also provided examples to illustrate the findings.