Data-driven deep learning (DL) techniques developed for automatic modulation classification (AMC) of wireless signals are vulnerable to adversarial attacks. This poses a severe security threat to the DL-based wireless systems, specifically for edge applications of AMC. In this work, we address the joint problem of developing optimized DL models that are also robust against adversarial attacks. This enables efficient and reliable deployment of DL-based AMC on edge devices. We first propose two optimized models using knowledge distillation and network pruning, followed by a computationally efficient adversarial training process to improve the robustness. Experimental results on five white-box attacks show that the proposed optimized and adversarially trained models can achieve better robustness than the standard (unoptimized) model. The two optimized models also achieve higher accuracy on clean (unattacked) samples, which is essential for the reliability of DL-based solutions at edge applications.