Yes, except $f(x)$ should be $1$ in the last line:$$b_n={2\over \pi}\int_0^\pi 1\cdot \sin(\sqrt{\lambda_n}\theta)\,d\theta={2\over \pi}\int_0^\pi \sin(n\theta)\,d\theta,\ n=1,2,\dots$$
Side note: good job eliminating the $r^\sqrt{\lambda_n}=r^n$ terms from your solution since they blow up as $r\to\infty$ in this "exterior (semi)disk" problem.