The deployment of O-RAN systems on general-purpose computing platforms represents a significant paradigm shift, promising remarkable performance improvements. However, these architectures may potentially increase both the capital and operational expenses of the network. The processor pooling concept is a promising solution to address this problem, consisting of a set of processing units (PUs) in the O-Cloud shared by several virtualized BSs (vBSs). Nevertheless, this strategy requires sophisticated resource assignment mechanisms to provide the expected gains in terms of cost and reliability. This paper proposes a novel online learning framework that assigns computing resources to vBSs in real-time (e.g., every TTI), thus handling the burstiness of real traffic loads. Our algorithm relies on online convex optimization (OCO) theory, extending state-of-the-art approaches in long-term fairness and constrained optimization and allowing discrete decisions. Our method offers an intrinsic closed-form iteration, speeding up the computation process and consequently allowing real-time operation. Moreover, our solution has guarantees in terms of fairness among the vBSs while adhering to long-term constraints in terms of energy over the entire operation horizon. We validate our theoretical findings via simulation and evaluate experimentally the algorithms in an O-RAN platform.