The question is to prove the mapping from $U(16)$ to itself given by $x \mapsto x^3$ is an automorphism.
It then asks to generalize for $x \mapsto x^5$ and $x \mapsto x^7$.
For the first part of the question, I listed the elements of $U(16)$ and where the mapping sends them showing that it's bijective and domain = codomain so it's operation preserving.
Is that enough to show something is an automorphism? Is there any better/ more efficient way to do this sort of question? Especially the generalizing portion, should I just compute where the 8 elements are sent to again?
Your method works, but consider this fact to show it's bijective:
for $x\in U(16)$, $x^8\equiv1\bmod 16$, so $(x^k)^k\equiv x^{k^2}\equiv x^1= x\bmod16,$ where $k\in\{3,5,7\}$
since $k^2\equiv1\pmod 8$.