  In this paper we deal with a particular class of rank two vector bundles
(\emph{instanton} bundles) on the Fano threefold of index one $F:=\mathbb{F}_1
\times \mathbb{P}^1$. We show that every instanton bundle on $F$ can be
described as the cohomology of a monad whose terms are free sheaves.
Furthermore we prove the existence of instanton bundles for any admissible
second Chern class and we construct a nice component of the moduli space where
they sit. Finally we show that minimal instanton bundles (i.e. with the least
possible degree of the second Chern class) are aCM and we describe their moduli
space.
