= Solution
Since $t=u+r$, the hypersurface $\Sigma_t$ is an ordinary constant-Minkowski-time slice. Its future unit normal is
$$
\boxed{n^a=(1,0,0,0)=\partial_u}
$$
in the $(u,r,\theta,\phi)$ basis. Lowering the index gives $n_a=(-1,-1,0,0)=-\nabla_at$. Restricting the metric to $dt=du+dr=0$ gives the Euclidean spherical metric
$$
dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2),
$$
so its induced <Riemannian volume form> is
$$
\boxed{d\sigma=r^2\sin\theta\,dr\,d\theta\,d\phi}.
$$
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