Let $S = (\{1,4,5,9,14,19,24\}, 19)$.
Let $T = (\{1,4,5,9,14,19,24\}, 1)$.
Define the morphism $f: S \to T$ as follows:
$$ f(n) = \left\{\begin{array}{lr} n , & \text{when } n \notin \{1,19\} \\ 1 , & \text{when } n = 19 \\ 19, & \text{when } n = 1 \end{array}\right\} $$
I think it is but I suspect that there is a better way of expressing $f$ using more 'category-centric ' notation. It isn't an automorphism, but does it have a name?
Why wouldn't it be?
In the category of pointed sets morphisms are simply functions $f: (X,x) \to (Y,y)$ where $f :X \to Y$ and $f(x) = y$. Your function is well defined and maps $x$ to $y$ indeed. Moreover, your morphism is a bijection and hence it is an isomorphism. Note that if $x \neq y$ then the objects $(X,x)$, $(X,y)$ are not equal. In fact you've shown they're isomorphic, though.