I am told that the diffusion equation for radially symmetric flow is $\frac{∂u}{∂t}=\frac{1}{r}\frac{∂}{∂r}\left(rD\frac{∂u}{∂r}\right)$. This is with flux $J=-D\frac{∂u}{∂r}$.
I can show that the steady state of this equation, with boundary conditions $u(a)=0$ and $u(b)=u_0$, is $u(r)=\frac{u_0}{log\left(\frac{b}{a}\right)}log\left(\frac{r}{a}\right)$. So for this steady state the flux is $J=-D\frac{∂u}{∂r}=-\frac{Du_0}{rlog\left(\frac{b}{a}\right)}$.
I am now asked to find the total amount of material in the region $a\leq r\leq b$ but I am confused how to do this, or what it has to do with the flux I just calculated.
Assuming that $u(r)$ refers to the concentration of the material in question, then all you have to do is integrate your expression for your steady-state $u(r)$ in the volume bounded by $a<r<b$.
The calculation has nothing to do with the flux of material exiting or entering the volume—the flux expression you find only tells you that those fluxes are the fluxes required to hold the concentration of material constant over time within that region.