Find a Möbius transformation such that a semi-circle in the upper half plane is mapped to $i\mathbb{R}_{>0}$

Your approach is fine. A Möbius transformation with $M(-2) = 0, M(2i) = i, M(2) = \infty$ maps the full circle $\{ |z|=2 \}$ to the extended line $i \Bbb R \cup \{ \infty \}$.

The semi-circle in the upper half-plane is then mapped to an open connected subset of that extended line having $0$ and $\infty$ as boundary points and containing the point $i$, and that is the “upper” part of the imaginary axis.

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