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.

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.