Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
The following sub question is rewritten thanks to the comment of Robert Bryant:
Is it true that if $(M,J)$ admits a vector field that preserves $J$ then there is $J'$ on $M$ homotopic to $J$ that is preserved by an $S^1$-action on $M$?
After three years I don't think indeed that the following is such a reasonable motivation
POSSIBLE MOTIVATION. Claire Voisin gave a construction of the Hilbert scheme of points for every almost complex 4-fold by an analogy with the Hilbert scheme of points of a complex surface. The first calculation of the Euler charactericstics of Hilbert scheme of points of complex surfaces was done via localisation techniques, for $CP^2$. Now, if we have a "holomorphic" vector field on an almost complex manifold, this could potentially help to reduce the calculation of the Euler charachteristics of its Hilbert scheme to the study of fixed points of the manifold. So the question is how flexible this notion is... But in its nature this question seems to be more a question (maybe not a hard one) on dynamical systems.