  In this article, we get classification theorems for a Ricci soliton on the
pseudo-Riemannian hypersurface of the Minkowski space $\mathbb{E}^4_1$ taking
the potential vector field as the tangent component of the position vector of
the pseudo-Riemannian hypersurface, denoted by $(M,g,{\bf x}^T,\lambda)$ in
both Riemannian and Lorentzian settings. First, we obtain the necessary and
sufficient condition that a pseudo-Riemannian hypersurface $(M,g)$ in
$\mathbb{E}^4_1$ admits a Ricci soliton $(M,g,{\bf x}^T,\lambda)$. In each of
the form of the shape operator of a pseudo-Riemannian hypersurface, we obtain
characterization a Ricci soliton on a pseudo-Riemannian hypersurface. More
precisely, we show that totally umbilical hypersurfaces, hyperbolic and a
pseudo-spherical cylinder in $\mathbb{E}^4_1$ is a shrinking Ricci soliton
whose the potential vector field is the tangent part of the position vector.
Furthermore, we conclude that there exists only a shrinking Ricci soliton on a
Lorentzian isoparametric hypersurface in $\mathbb{E}^4_1$ with
nondiagonalizable shape operator whose the minimal polynomial has double real
roots.
