The strong Kantorovich duality, i.e. the existence of dual solutions, holds whenever $c$ is lower semicontinuous and real-valued, and the optimal coupling has finite cost, see Theorem 5.10(ii) in Villani "Optimal Transport old and new". In contrast, for the weak duality ("$\inf = \sup$") it suffices to assume $c$ is lower semicontinuous.
To prove 5.10(ii) one constructs an explicit $c$-concave function $\varphi$ that is finite-valued on the support of $\mu$, see formula (5.17). Beware of signs! Villani is looking at $c$-convex functions and you at $c$-concave functions.
Note if $c$ is not real-valued then there may not be any dual solutions between certain measures $\mu$ and $\nu$, see the preprint of S. Suhr and me on Lorentzian cost function, more explicitely Section 3.1.