  Let $X \subset \mathbb P^3$ be a very general sextic surface over complex
numbers. In this paper we study certain Brill-Noether problems for moduli of
rank $2$ stable bundles on $X$ and its relation with rank $2$ weakly Ulrich and
Ulrich bundles. In particular, we show the non-emptiness of certain
Brill-Noether loci and using the geometry of the moduli and the notion of the
Petri map on higher dimensional varieties, we prove the existence of components
of expected dimension. We also give sufficient conditions for the existence of
rank $2$ weakly Ulrich bundles $\mathcal E$ on $X$ with $c_1(\mathcal E) =5H$
and $c_2 \geq 91$ and partially address the question of whether these
conditions really hold. We then study the possible implication of the existence
of an weakly Ulrich bundle in terms of non-emptiness of Brill-Noether loci.
Finally, using the existence of rank $2$ Ulrich bundles on $X$ we obtain some
more non-empty Brill-Noether loci and investigate the possibility of existence
of higher rank simple Ulrich bundles on $X$.
