This paper integrates non-orthogonal multiple access (NOMA) and over-the-air federated learning (AirFL) into a unified framework using one simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS). The STAR-RIS plays an important role in adjusting the decoding order of hybrid users for efficient interference mitigation and omni-directional coverage extension. To capture the impact of non-ideal wireless channels on AirFL, a closed-form expression for the optimality gap (a.k.a. convergence upper bound) between the actual loss and the optimal loss is derived. This analysis reveals that the learning performance is significantly affected by the active and passive beamforming schemes as well as wireless noise. Furthermore, when the learning rate diminishes as the training proceeds, the optimality gap is explicitly shown to converge with linear rate. To accelerate convergence while satisfying quality-of-service requirements, a mixed-integer non-linear programming (MINLP) problem is formulated by jointly designing the transmit power at users and the configuration mode of STAR-RIS. Next, a trust region-based successive convex approximation method and a penalty-based semidefinite relaxation approach are proposed to handle the decoupled non-convex subproblems iteratively. An alternating optimization algorithm is then developed to find a suboptimal solution for the original MINLP problem. Extensive simulation results show that i) the proposed framework can efficiently support NOMA and AirFL users via concurrent uplink communications, ii) our algorithms achieve faster convergence rate on IID and non-IID settings compared to existing baselines, and iii) both the spectrum efficiency and learning performance is significantly improved with the aid of the well-tuned STAR-RIS.