Abstract: Neutral atom quantum processors are emerging as a leading platform for quantum computing. Integrating them with an optical cavity enables new capabilities, such as fast nondestructive readout, all-to-all connectivity, and access to fast atom-photon entanglement for quantum networking. Here, we introduce a platform that transports and couples single atoms in optical tweezers to a Fabry-Perot Fiber Cavity. Leveraging the strong atom-cavity coupling, we demonstrate fast qubit state readout with 99.96(2) % fidelity in 10μs and two methods for cavity-mediated entanglement generation with integrated error detection. First, we use cavity-carving to generate a Bell state with fidelity 91(4)% and with 32(1)% success rate. Second, we perform a cavity-mediated gate with a deterministic entanglement fidelity of 52.5(18)%, increased to 76(2)% with error detection. The new capabilities enabled by this versatile platform pave the way towards modular quantum computing.