Indeed, Runge's theorem is just the thing for this problem. Given a compact subset $K\subset V$ and a number $\epsilon>0$, we must show that for every $f\in \mathcal O(U)$ there exists $g\in\mathcal O(V)$ such that $\sup_K|f-g|<\epsilon$. We want $g$ to be a rational function with poles outside of $V$. According to Runge, we must choose a point in each connected component of $\overline{\mathbb C}\setminus K$ where the poles of $g$ may be located. Let us be careful here: we don't want $K$ to have extra holes in it. By enlarging $K$ we can make sure that each connected component of $\overline{\mathbb C}\setminus K$ contains a component of $\overline{\mathbb C}\setminus U$. One way to do it is to let $K$ consist of all points of $U$ whose spherical distance to $\overline{\mathbb C}\setminus U$ is at least $\epsilon$.
It remains to prove that no connected component of $\overline{\mathbb C}\setminus K$ is contained in $V$. Suppose, to the contrary, that $\Omega$ is a connected component of $\overline{\mathbb C}\setminus K$ that is contained in $V$. Let $E$ be a connected component of $\overline{\mathbb C}\setminus U$ that is contained in $\Omega$. Being a connected component, $E$ is closed in $\overline{\mathbb C}\setminus U$. And since $\overline{\mathbb C}\setminus U$ is a compact set, so is $E$. Finally, $E$ is also a component of $V\setminus U$ because $E\subset V$. A contradiction.