The problem is
"Find a Möbius transformation $w(z)$ that maps the area $Re( z) > 0, |z-1| > 1$ to the strip $0<Re(w)<2$."
I realize there are many ways to skin a cat, but what I wanted to do was map $0\mapsto \infty$ and then conclude that since the mirror point of $z=0$ with respect to $|z-1| = 1$ is 2, then $w(z)$ must map 2 to the mirror point of $w=\infty$ with respect to one of the boundaries.
But I'm not really sure what the mirror point of $w=\infty$ is. A reasonable guess, in my mind, is $\infty$ but I don't think that's correct. Can anyone explain?
I'm not sure what your mirroring is supposed to do but one way of finding the map is to simply realize that $0$ must be mapped to $\infty$ since $0$ is the intersection of the boundary line and circle and $\infty$ the intersection of the two boundary lines after transformation. $\infty$ must then be mapped to a point lying on imaginary axis, so by symmetry let's pick $0$. Similarly, $2$ must be mapped to the vertical line $2+i{\bf R}$, so again by symmetry let us take $2$. Therefore $w(z) = 4/z$ does the job (check that this maps boundaries to boundaries, as it should).