The energy-momentum tensor of a perfect fluid is $$T_{\mu\nu}=(p+\rho)u_\mu u_\nu + pg_{\mu\nu}$$ The local conservation of energy and momentum is expressed by the equation $$\nabla_\mu T^{\mu\nu}=(p+\rho)u^\mu\nabla_\mu u^\nu+g^{\mu\nu}\partial_\mu p+u^\nu\left[\nabla_\mu(p+\rho)u^\mu\right]=0$$
By contracting with the projection tensor $h_{\mu\nu}=g_{\mu\nu}+u_\mu u_\nu$, we can obtain the relativistic Euler equation $$(p+\rho)u^\mu\nabla_\mu u^\nu + (g^{\mu\nu}+u^\mu u^\nu)\partial_\mu p=0$$
This is the equation of motion for fluid flows in a background metric $g_{\mu\nu}$. On the other hand, the motion of particles is governed by the geodesic equation.
Since hydrodynamics is the continuum limit of particle dynamics, is it possible to derive the geodesic equation from the relativistic Euler equation as a limiting case?
The "particles" in a relativistic prefect fluid do not necessarily follow geodesics; the pressure essentially arises from short-ranged interactions between the particles. Another way of seeing this is that the nonrelativistic limit is a classical ideal fluid, which certainly does not always flow in straight lines.
If pressure is zero, however (i.e. "perfect dust"), then particles do follow geodesics. Is it evident from the relativistic Euler equation you have written that this is the case?
(recall the form of the geodesic equation $\nabla_{\dot{\gamma}} \dot{\gamma}=0$)