If $M$ is a $4$-manifold, the existence of an almost-complex structure can be often detected by using the following result due to Wu:
Theorem. A $4$-manifold $M$ admits an almost-complex structure $J$ if and only if there exists $h \in H^2(M, \mathbb{Z})$ such that $$h^2=3 \sigma(X)+2 \chi(X) \quad \textrm{and} \quad h \equiv w_2(X) \; \; \textrm{mod} \ 2. $$ In this case $h=c_1(M, J)$.
See Gompf-Stipsicz, 4-manifolds and Kirby calculus, p. 30 for more details.