We prove the equivalence between the notion of Wasserstein gradient flow for
a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian
potential on one side, and the notion of entropy solution of a Burgers-type
scalar conservation law on the other. The solution of the former is obtained by
spatially differentiating the solution of the latter. The proof uses an
intermediate step, namely the L2 gradient flow of the pseudo-inverse
distribution function of the gradient flow solution. We use this equivalence to
provide a rigorous particle-system approximation to the Wasserstein gradient
flow, avoiding the regularization effect due to the singularity in the
repulsive kernel. The abstract particle method relies on the so-called
wave-front-tracking algorithm for scalar conservation laws. Finally, we provide
a characterization of the sub-differential of the functional involved in the
Wasserstein gradient flow