A regular language derived from another
In order to show that $f$ is regularity preserving, it suffices to show that $f^{-1}(U)$ is eventually periodic for $U = \{ n : n \equiv b \pmod{a} \}$, where $a$ is a prime power (here we are using the Chinese remainder theorem and the fact that the eventually periodic sets are closed under intersection).
For $f(n) = 2^n$, we consider two cases:

If $a = 2^k$ then $f(n) \equiv 0 \pmod{a}$ for large enough $n$.

If $a = p^k$ for an odd prime $p$, then $(2,a) = 1$, and so we can think of $2$ as an element of the multiplicative group $\mathbb{Z}_a^\times$. In particular, $2^n \bmod a$ is periodic, the period being the order of $2$ in the group.