With the aim of integrating over-the-air federated learning (AirFL) and non-orthogonal multiple access (NOMA) into an on-demand universal framework, this paper proposes a novel reconfigurable intelligent surface (RIS)-aided hybrid network by leveraging the RIS to flexibly adjust the signal processing order of heterogeneous data. The objective of this work is to maximize the achievable hybrid rate by jointly optimizing the transmit power, controlling the receive scalar, and designing the phase shifts. Since the concurrent transmissions of all computation and communication signals are aided by the discrete phase shifts at the RIS, the considered problem (P0) is a challenging mixed integer programming problem. To tackle this intractable issue, we decompose the original problem (P0) into a non-convex problem (P1) and a combinatorial problem (P2), which are characterized by the continuous and discrete variables, respectively. For the transceiver design problem (P1), the power allocation subproblem is first solved by invoking the difference-of-convex programming, and then the receive control subproblem is addressed by using the successive convex approximation, where the closed-form expressions of simplified cases are derived to obtain deep insights. For the reflection design problem (P2), the relaxation-then-quantization method is adopted to find a suboptimal solution for striking a trade-off between complexity and performance. Afterwards, an alternating optimization algorithm is developed to solve the non-linear and non-convex problem (P0) iteratively. Finally, simulation results reveal that 1) the proposed RIS-aided hybrid network can support the on-demand communication and computation efficiently, 2) the performance gains can be improved by properly selecting the location of the RIS, and 3) the designed algorithms are also applicable to conventional networks with only AirFL or NOMA users.