We consider $\alpha$-almost Ricci solitons on $(k,\mu)'$-almost Kenmotsu manifolds with an $\eta$-parallel Ricci tensor. Then we study $\alpha$-almost Ricci solitons on $(k,\mu)'$-almost Kenmotsu manifolds satisfying the curvature conditions $P.\phi = 0$, $Q.P = 0$ and $Q.R = 0$ respectively. Finally, we construct an example of a 3-dimensional $(k,\mu)'$-almost Kenmotsu manifold which admits an $\alpha$-almost Ricci soliton.