I think this can be done by contradiction. Suppose not. Then $\forall f \in \mathbb{Z}_n[x]$ of degree $1$ there does not exist $g \in \mathbb{Z}_n[x]$ such that $fg=1$, but I don't know where to go from here.
This might not be the best approach though. I don't understand what $U(R[x])) = U(R)$ means because isn't $U(R)$ all polynomials of degree $0$? Can someone please explain this and give me a hint?