In this paper, we investigate the performance of a fluid antenna relay (FAR)-assisted downlink communication system utilizing non-orthogonal multiple access (NOMA). The FAR, which integrates a fluid antenna system (FAS), is equipped on an autonomous aerial vehicle (AAV), and introduces extra degrees of freedom to improve the performance of the system. The transmission is divided into a first phase from the base station (BS) to the users and the FAR, and a second phase where the FAR forwards the signal using amplify-and-forward (AF) or decode-and-forward (DF) relaying to reduce the outage probability (OP) for the user maintaining weaker channel conditions. To analyze the OP performance of the weak user, Copula theory and the Gaussian copula function are employed to model the statistical distribution of the FAS channels. Analytical expressions for weak user's OP are derived for both the AF and the DF schemes. Simulation results validate the effectiveness of the proposed scheme, showing that it consistently outperforms benchmark schemes without the FAR. In addition, numerical simulations also demonstrate the values of the relaying scheme selection parameter under different FAR positions and communication outage thresholds.