Sounds like a home work; here are some hints:
The capacity is equal to the integral of |gradient|$^2$ of spherically symmetric harmonic function with 1 on the bry of the ball and zero at infinity. The function $f$ depends only on the radius, say $r$. You can cook an ODE for $f$, something like $$f''(r)+\frac{(n-1){\cdot}f'(r)}{\sinh r}=0.$$ $f''(r)+\frac{(n-1){\cdot}\cosh r}{\sinh r}{\cdot}f'(r)=0.$$ Then you should solve it and integrate $$\mathrm{vol}\, S^{n-1}{\cdot}\int\limits_R^\infty(f')^2{\cdot}(\sinh r)^{n-1}\, dr.$$
(I do not know if will get the answer in a simple form, but it will be good for all practical purposes...)
