In this paper, the authors study approximate necessary optimality conditions for weakly $\epsilon$-efficient solutions to robust multi-objective convex optimization problems, by dealing with the associated minimax optimization problem and under the robust characteristic cone constraint qualification, which is weaker than the Slater condition. Technically, the optimality result is established by employing the $\alpha$-subdifferential of the max-function and the variational representation of the $\beta$-normal set.