I think you also want $\lim_{r \to +\infty} f(r)$ and $\lim_{r \to -\infty} f(r)$ to exist and be equal. Schwarz Reflection principle shows $f$ is meromorphic on $\mathbb C$ with $f(\overline{z}) = 1/\overline{f(z)}$. Same applies to $f(1/z)$. So $f$ is an analytic function from the Riemann sphere to itself, and such functions are rational.
As far as I can tell this involves little to no complex analysis. Also, I'm assuming you want $S$ to be open in the upper half plane, not necessarily in all of $\mathbb{C}$.
To construct $S$, consider any $[a,b] \times [0, R] \subseteq \mathbb{H}$ (that is, $\{z: a \leqslant Re z \leqslant b, 0 \leqslant Im z \leqslant R \}$, for $R >> 0$, by compactness and hence uniform continuity there exists a subset $S_{a,b}$ of the form $[a,b] \times [0, \epsilon)$ on which $f$ is nonvanishing.
Now taking $[a,b]$ to be say $\ldots [-1,0], [0,1], [1,2], \ldots$, and patching your $S_{a,b}$ together nicely at the endpoints gives $S$.
$S$ is clearly simply connected (it deformation retracts onto the real line via "squishing").
Lastly, to construct $\psi$, we have $\Omega := S - \mathbb{R}$ is open and simply connected, $f$ is nonvanishing on $\Omega$, and hence $f = e^{g(z)}$ for $g$ holomorphic. Take $\psi := g/i$.
If this is unclear (or wrong!) lemme know.
Best Answer
Your idea is correct: You start by defining the extended function $$\tilde{f}:\mathbb{C}\to \mathbb{C},\quad \tilde{f}(z):=\begin{cases} f(z) & z\in\overline{\mathbb{H}}\\ \overline{f(z)} & z\in \mathbb{C}\setminus \overline{\mathbb{H}} \end{cases}.$$ By the Schwarz reflection principle this function is holomorphic on the whole of $\mathbb{C}$, i.e. entire.
But this entire function again satisfies $\Re(\tilde{f})\ge 0$. Now the function $$z\mapsto e^{-\tilde{f}(z)}$$ is again entire and is bounded, as $$|e^{-\tilde{f}(z)}|=|e^{-\Re(\tilde{f}(z))}|\leq 1.$$ Thus by Liouville $e^{-\tilde{f}(z)}$ is constant. But this implies that $\tilde{f}(z)$ is constant and hence $f$ itself is constant.