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)-unified network by leveraging the RIS to flexibly adjust the signal processing order of hybrid users. The objective of this work is to maximize the achievable hybrid rate of all users by jointly optimizing the transmit power at users, controlling the receive scalar at the base station, and designing the phase shifts at the RIS. Due to the fact that all signals of computation and communication are combined into one concurrent transmission, the formulated resource allocation problem is very challenging. To solve this problem, the alternating optimization is invoked to address non-convex subproblems iteratively for finding a suboptimal solution with low complexity. Simulation results demonstrate that i) the proposed RIS-unified network can support the on-demand communication and computation efficiently, and ii) the designed algorithms are also applicable to conventional networks with only AirFL or NOMA users.