  Let $\mathcal{E}$ be a vector bundle on a smooth projective variety
$X\subseteq\mathbb{P}^N$ that is Ulrich with respect to the hyperplane section
$H$. In this article, we study the Koszul property of $\mathcal{E}$, the
slope-semistability of the $k$-th iterated syzygy bundle
$\mathcal{S}_k(\mathcal{E})$ for all $k\geq 0$ and rationality of moduli spaces
of slope-stable bundles on Del Pezzo surfaces. As a consequence of our study,
we show that if $X$ is a Del Pezzo surface of degree $d\geq 4$, then any Ulrich
bundle $\mathcal{E}$ satisfies the Koszul property and is slope-semistable. We
also show that, for infinitely many Chern characters ${\bf v}=(r,c_1, c_2)$,
the corresponding moduli spaces of slope-stable bundles $\mathfrak{M}_H({\bf
v})$ when non-empty, are rational, and thereby produce new evidences for a
conjecture of Costa and Mir\'o-Roig. As a consequence, we show that the
iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.
