We study the torus-equivariant homology $H_*^T(\mathrm{Gr}_G)$ of the affine Grassmannian $\mathrm{Gr}_G$, where $G=\mathrm{Sp}_{2n}(\mathbb{C})$ is the symplectic group. This homology admits a natural ring structure and a Schubert basis, giving rise to a well-defined Schubert calculus. We realize $H_*^T(\mathrm{Gr}_G)$ in terms of symmetric functions. Our first main result introduces a new family of symmetric functions, called the \emph{dual affine Schur $P$-functions}, which represent the Schubert classes. These functions are defined through the action of the affine nil-Hecke algebra, and specialize, in the stable limit as $n\to \infty$, to the dual factorial $P$-functions of Nakagawa and Naruse. Our second main result gives a precise comparison between this symmetric function model and the geometric construction of $H_*^T (\mathrm{Gr}_G)$ due to Ginzburg and Peterson, which identifies it with a coordinate ring of a centralizer family in the Langlands dual group.