What are Ring automorphism of $\mathbb{Z}_4 ?$ As per the theory, since it's a cyclic group, it depends on the image of the generating element 1.
so there are two possiblities, $\phi(1)=1, \phi(1)=3$
But the isomorphic property says, the map should take identity to identity i.e $\phi(1)=1.$
So in that case $\phi(1)=3$ becomes invalid. Am I correct?
You are mixing thoughts about group isomorphisms and ring isomorphisms.
When you think of the map $1 \mapsto 3$, you must have $1^2 \mapsto 3^2 = 1$, but $1 = 1^2$, so in the image, we are forced to have $3 = 1$, which is not true in $\Bbb{Z}/4\Bbb{Z}$.
What you have written about having two isomorphisms is correct for the abelian group of integers modulo $4$ under addition. As just shown, it is not the case that both maps preserve the multiplication, essentially for the ring isomorphism reason you mention: the multiplicative identity must be mapped to the multiplicative identity.