= Solution
For each of the two normal directions, launch a <null geodesic congruence> orthogonally from the sphere with initial tangent $U=X$ or $U=Y$. For $U=X$ take initial auxiliary $N=Y$; for $U=Y$ take initial $N=X$, and then <parallel transport> $N$ as in part (a). On the sphere the resulting <screen-space projector> projects onto its angular tangent space, whose metric is $q_{AB}=r^2\gamma_{AB}$.
For any radial normal $K$, the <spherical null expansion in ingoing coordinates> follows directly from area variation:
$$
\theta_K=\frac12q^{AB}\mathcal L_Kq_{AB}=\frac{2K(r)}r.
$$
Equivalently $\mathcal A\propto r^2$ gives $K(\log\mathcal A)=2K(r)/r$. Consequently
$$
\boxed{\theta_X=-\frac2{r_0},\qquad \theta_Y=\frac{f(r_0)}{r_0}=\frac{(r_0-r_+)(r_0-r_-)}{r_0^3}.}
$$
Spherical symmetry also makes the <null shear> and <null twist> zero for these radial congruences.
There is a small parametrization distinction. The smooth field $X=-\partial_r$ is affine, since $\Gamma^a{}_{rr}=0$. The natural smooth field $Y=\partial_v+(f/2)\partial_r$ obeys $\nabla_Y Y=(f'/2)Y$. To meet part (a)'s affine convention, use an <affine rescaling of a null normal> along the outgoing generators, with scaling equal to one on $S$. Its angular derivatives on $S$ then leave the projected derivative unchanged. The displayed <null expansions> are therefore precisely those for the affinely launched congruences, even though that convenient global expression for $Y$ is nonaffine away from its initial sphere.
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