This is a good exercise. Let $\phi$ be the conformal mapping of the half plane to the unit disk.
To create a harmonic function on $\mathbb{H}$ which agrees with $f$ on the real line, one good strategy would be to translate it to the unit disk. Using the Poisson kernel for the disk, we can find a harmonic function on the disk which agrees with $f\circ \phi^{-1}$ on the boundary. Compose it with $\phi$ (which is also harmonic) to get a function which is harmonic on $\mathbb{H}$ that agrees with $f$ on the real line.
This is an outline, in the sense that to derive the Poisson kernel for the upper half plane, you have to power through some algebraic manipulations. That is messy, but not hard (especially if you know the answer).
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
There are Cauchy's estimates for derivatives of analytic functions: if $f$ is holomorphic in a neighourhood of the closed ball $\bar{B}(z_0)$, then, for every $z\in B(z_0)$ you have an estimate for the $k$-th derivative, $$ |f^{(k)}(z)| \le k!\frac{r}{d(z)^{k+1}}|f|_{\partial B}, \,\, d(z)= \mbox{dist}(z, \partial B(z_0))$$ You can apply this with $z_0=i$, any ball around $i$ of radius less than $1$ and the knowledge that $|f|\le 1$ (this is, in particuar, a bound for $f$ at the boundary of the ball).