What's the proof that the only bijective group automorphism (or isomorphism) in $\mathbb{Z}_n$ is in form $\psi(n)=\psi(1)\cdot n$, where $\psi(n)\in\mathbb{Z}_n$?
If it so, can be this generalized to any finite group (e.g. $\psi(n)=(\psi(1))^n$ for group $S_n$)?
It's not unique. Since $1$ and $2$ both generate $\mathbb Z/3\mathbb Z$, the identity map and the map sending $1\mapsto 2$, $2\mapsto 1$ and $0\mapsto 0$, you can check, are both automorphisms, and they are different.
However, both of these maps are uniquely determined by the image of $1$. Perhaps this is what you should be checking: that any automorphism of a cyclic group is determined by the image of a generator.