This paper deals with the validation of convergence results for the newly proposed $C$-$\alpha$ non-expansive mappings in the CAT$(0)$ space setting. Before analyzing the convergence behavior, we emphasize key results related to $C$-$\alpha$ non-expansive mappings. The convergence results are obtained using the $JF$-iteration method. We then illustrate these results with non-trivial examples and compare them to other notable iterations. These comparisons are presented in both tabular and graphical forms. Finally, we discuss a variational inequality problem within the context of these mappings using the same iteration scheme.