According to Mukai and Iliev, a smooth prime Fano threefold \(X\) of genus \(9\) is associated with a surface \(\mathbb{P }(\mathcal{V })\), ruled over a smooth plane quartic \(\varGamma \), and the derived category of \(\varGamma \) embeds into that of \(X\) by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank-\(2\) stable sheaves on \(X\) with odd determinant. For each \(c_2 \ge 7\), we prove that a component of their moduli space \(\mathsf{M}_X(2,1,c_2)\) is birational to a Brill–Noether locus of vector bundles with fixed rank and degree on \(\varGamma \), having enough sections when twisted by \(\mathcal{V }\). For \(c_2=7\), we prove that \(\mathsf{M}_X(2,1,7)\) is isomorphic to the blow-up of the Picard variety \(\text{ Pic}^{2}({\varGamma })\) along the curve parametrizing lines contained in \(X\).