Artificial Intelligence (AI)-powered Radio Access Network (RAN) networks have attracted significant attention from both industry and academia. Meanwhile, Digital Twins offer a safe playground for experimenting with AI/Machine Learning (ML)-based solutions for advanced AI-RAN research. By enabling the testing of online algorithms before deployment on the RAN, they reduce costs and safety risks associated with physical field testing. In this article, we propose ARIADNE, an online Reinforcement Learning (RL)-based module that seamlessly integrates with SIONNA and is tasked with performing link adaptation. We explore different design choices and demonstrate how ARIADNE can surpass industry-standard and state-of-the-art methods by achieving up to 11% and 20% improvements in Spectral Efficiency, respectively. Finally, we show that RL learns a Modulation and Coding Scheme (MCS) selection strategy that diverges from Outer Loop Link Adaptation (OLLA), exhibiting either more conservative or more aggressive behavior depending on the configuration, a trend further corroborated by training offline on 5th generation (5G) over-the-air (OTA) measurements.