How to get Euler-Lagrange equations out?

I'll expand on @QMechanic's answer. The first equation is straightforward. For the second, note your action is$$S=\int d^4x\mathcal{L},\,\mathcal{L}:=\frac12\rho v_iv_i-u(\rho)+\phi\dot{\rho}+\phi\rho\partial_iv_i+\phi v_i\partial_i\rho,$$so$$0=\frac{\partial\mathcal{L}}{\partial v_i}-\partial_j\frac{\partial\mathcal{L}}{\partial\partial_jv_i}=\rho v_i+\phi \partial_i\rho-\partial_j\left(\phi\rho\delta_{ij}\right)=\rho v_i-\rho\partial_i\phi=\rho\left(v-\nabla\phi\right)_i,$$which reduces to the given result. Finally, varying $\rho$ gives$$0=\frac{\partial\mathcal{L}}{\partial\rho}-\partial_j\frac{\partial\mathcal{L}}{\partial\partial_j\rho}-\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\rho}}=\frac12 v^2-u^\prime+\phi\nabla\cdot v-\partial_j\left(\phi v_j\right)-\dot{\phi}\\=\frac12 v^2-u^\prime-v\nabla\phi-\dot{\phi},$$which again is what was claimed.

Incidentally, a physical interpretation of these equations provides a sanity check: @QMechanic's first result was mass conservation, the second identified $v$ with $\phi$'s gradient as we expected, and third is Navier-Stokes and analogous to Newton's second law.