  The aim of this paper is to continue the study of Kodaira dimension for
almost complex manifolds, focusing on the case of compact $4$-dimensional
solvmanifolds without any integrable almost complex structure. According to the
classification theory we consider: $\mathfrak{r}\mathfrak{r}_{3, -1}$,
$\mathfrak{nil}^4$ and $\mathfrak{r}_{4, \lambda, -(1 + \lambda)}$ with $-1 <
\lambda < -\frac{1}{2}$. For the first solvmanifold we introduce special
families of almost complex structures, compute the corresponding Kodaira
dimension and show that it is no longer a deformation invariant. Moreover we
prove Ricci flatness of the canonical connection for the almost K\"ahler
structure. Regarding the other two manifolds we compute the Kodaira dimension
for certain almost complex structures. Finally we construct a natural
hypercomplex structure providing a twistorial description.
