For the domain \begin{equation} E_r=\{z: r < |z| < 1\}, \end{equation} find a harmonic function $ u $ in $ E_r $ so that \begin{equation} u(re^{i\theta})=1 \text{ and } u(e^{i\theta})=0 \end{equation}
I am trying to figure out a good conformal map, but the problem is, the region is not simply connected, so I cannot map it on the unit disk or a half-plane
I am not sure why you need to find a conformal map, but if you simply want a harmonic function in $E_r=\{z: r < |z| < 1\}$ which satisfies your conditions on the boundary, then you can find one as follows:
The function $u(z) = a+b\log |z|$ is well-known to be harmonic (and can easily be checked) in $\mathbb{C}\setminus \{0\}$, and choosing $a = 0$ and $b = \frac{1}{\log r}$, will give you the function $u(z) = \frac{\log |z|}{\log r} $ which is harmonic in $E_r$ and has the boundary values you require.