  In this work we characterize Ulrich bundles of any rank on polarized rational
ruled surfaces over $\mathbb{P}^1$. We show that every Ulrich bundle admits a
resolution in terms of line bundles. Conversely, given an injective map between
suitable totally decomposed vector bundles, we show that its cokernel is Ulrich
if it satisfies a vanishing in cohomology. As a consequence we obtain, once we
fix a polarization, the existence of Ulrich bundles for any admissible rank and
first Chern class. Moreover we show the existence of stable Ulrich bundles for
certain pairs $(\textrm{rk}(E),c_1(E))$ and with respect to a family of
polarizations. Finally we construct examples of indecomposable Ulrich bundles
for several different polarizations and ranks.
