In this paper, we investigate the distributed approximate subgradient-type method for minimizing a sum of differentiable and non-differentiable convex functions subject to nondifferentiable convex functional constraints in a Euclidean space. We establish the convergence of the sequence generated by our method to an optimal solution of the problem under consideration. Moreover, we derive a convergence rate of order $\mathcal{O}(N^{1-a})$ for the objective function values, where $a\in (0.5,1)$. Finally, we provide a numerical example illustrating the effectiveness of the proposed method.