Both questions have exact explicit answer, which is explained in any good textbook on analytic functions. A region on the Riemann sphere is called a digon if its boundary consists of two arcs of circles. Let $a$ and $b$ be the common endpoints of these arcs. Then $$f(z)=\frac{z-a}{z-b}$$ sends $a,b$ to $0,\infty$, and the arcs to rays. So the image of our region is a sector. The interior angle at a corner of the digon be $\pi\alpha$. Then the interior angle at the vertex of the sector is also $\pi\alpha$. Next $f_1(z)=z^{1/\alpha}$ maps this sector onto a half-plane. And the final ingredient is conformal map of this half-plane onto the unit disk.
EDIT 2. Asymptotics is $\mathrm{cap}(A_r)=1+r^2/2+O(r^3),$ as $r\to 0$. The exact formula is a bit complicated: $$\frac{1}{\mathrm{cap}(A_r)}=\frac{1}{4b}(1+\phi)(1+b^2)\sin\left(\frac{4}{1+\phi}\arctan b\right),$$ where $$\phi=\frac{2}{\pi}\arcsin(r/2),$$ and $$b=\frac{r}{2\sqrt{1-r^2/4}}.$$