Using scale factors

The length scale factor between the smaller cylinder and the larger cylinder is 2, so the volume scale fator between the smaller cylinder and the larger cylinder is 2$^3$=8.

So the smaller cylinder will contain $\frac{1}{9}$ of the total volume, and the larger cylinder will contain $\frac{8}{9}$ of the total volume.

The volume of the cube is $9^3$ cm$^3$, so the volume of the smaller cylinder is $\frac{1}{9}\times9^3=81$ cm$^3$.

Using expressions for the volumes of the two cylinders

Suppose the smaller cylinder has height $h$ cm, so the larger cylinder has height $2h$ cm. Then the volume of the smaller cylinder is $\pi\times2^2\times h=4\pi h$ cm$^3$ and the volume of the larger cylinder is $\pi\times4^2\times2h=32\pi h$ cm$^3$.

The volumes of the cylinders will have to add up to the volume of the cube, so $$\begin{align}4\pi h+32\pi h=&9^3\\

\Rightarrow 36\pi h=&729\\

\Rightarrow h=&\frac{729}{36\pi}=\frac{81}{4\pi}\end{align}$$

Substituting into the expression for the volume of the smaller cylinder gives $4\pi h=4\pi\times\dfrac{81}{4\pi}=81$. So the volume of the smaller cylinder is $81$ cm$^3$.