  We study the spin Gromov-Witten (GW) theory of $\mathbb{P}^1$. Using the
standard torus action on $\mathbb{P}^1$, we prove that the associated
equivariant potential can be expressed by means of operator formalism and
satisfies the 2-BKP hierarchy. As a consequence of this result, we prove the
spin analogue of the GW/Hurwitz correspondence of Okounkov-Pandharipande for
$\mathbb{P}^1$, which was conjectured by J. Lee. Finally, we prove that this
correspondence for a general target spin curve follows from a conjectural
degeneration formula for spin GW invariants that holds in virtual dimension 0.
