The reason I didn't mention hyperbolic geometry specifically is that vector calculus and linear algebra prepare one for a basis course on the differential geometry of curves and surfaces in 3-space. This is just what one would need to understand hyperbolic geometry in the right context (even though a complete hyperbolic plane of constant Gaussian curvature = -1 cannot be embedded in 3-space if it is of differentiability class C2. — Hilbert's theorem).
Hyperbolic geometry is ideally learned as an advanced topic in a course on the differential geometry of curves and surfaces in 3-space. So, the prerequisites are for such a course.