In this paper, we consider a downlink wireless communication system with the co-existence of ground user (GU) and mobile aerial user (AU). Existing solutions rely on orthogonal multiple access (OMA) to support these users, however, OMA is unable to provide the best rate and outage performance because its spectral efficiency is limited by the users’ channel conditions and rate requirements. Thus, we propose an aerial-ground non-orthogonal multiple access (AG-NOMA) scheme that pairs the GU and AU for data and control links, respectively. Unlike terrestrial non-orthgonal multiple access (NOMA), the key idea of AG-NOMA is to exploit the asymmetric features of the channels and rate demands of the GU and AU in the downlink communication. Based on these opportunities, we investigate the maximum achievable GU rate over a time-varying wireless channel while satisfying the AU Quality-of-Service (QoS) requirement with perfect and partial channel state information (CSI). For perfect CSI, we derive the optimal successive interference cancellation (SIC) policy, power allocation, GU rate, and feasibility conditions in closed-form expressions. For partial CSI, we also derive the suboptimal SIC policy and power allocation in closed-form expressions, and further discussed a tradeoff between the achievable rate and reliability. This tradeoff depends on the system parameters, and thus we have suggested some appropriate parameters based on theoretical support and standard requirements to strike a balance between rate and reliability. Our simulation results show that AG-NOMA scheme with perfect and partial CSI can achieve up to +99% GU rate-improvement as compared to OMA and provide a more sustainable rate-improvement and/or lower outage probability than terrestrial NOMA scheme.