Reinforcement learning (RL) has recently proven itself as a powerful instrument for solving complex problems and even surpassed human performance in several challenging applications. This signifies that RL algorithms can be used in the autonomous air combat problem, which has been studied for many years. The complexity of air combat arises from aggressive close-range maneuvers and agile enemy behaviors. In addition to these complexities, there may be uncertainties in real-life scenarios due to sensor errors, which prevent estimation of the actual position of the enemy. In this case, autonomous aircraft should be successful even in the noisy environments. In this study, we developed an air combat simulation, which provides noisy observations to the agents, therefore, make the air combat problem even more challenging. Thus, we present a state stacking method for noisy RL environments as a noise reduction technique. In our extensive set of experiments, the proposed method significantly outperforms the baseline algorithms in terms of the winning ratio, where the performance improvement is even more pronounced in the high noise levels. In addition, we incorporate a self-play scheme to our training process by periodically updating the enemy with a frozen copy of the training agent. By this way, the training agent performs air combat simulations to an enemy with smarter strategies, which improves the performance and robustness of the agents. In our simulations, we demonstrate that the self-play scheme provides important performance gains compared to the classical RL training.