Using there hint seek a solution in the form
$\displaystyle
u(r,\theta) = \sum_{n=1}^{\infty} R_n(r)\sin 2 n \theta
$.
When you sub in your PDE you obtain
$\displaystyle
\sum_{n=1}^{\infty} \left( R_n'' + \frac{R_n'}{r} - \frac{4n^2 R_n}{r^2}\right) \sin 2 n \theta = r \sin 2 \theta.
$
Now start indexing n and compare LHS to RHS. You'll get a set of ODEs for $\displaystyle R_n$.