Underwater Acoustic (UWA) networks are vital for remote sensing and ocean exploration but face inherent challenges such as limited bandwidth, long propagation delays, and highly dynamic channels. These constraints hinder real-time communication and degrade overall system performance. To address these challenges, this paper proposes a bilevel Multi-Armed Bandit (MAB) framework. At the fast inner level, a Contextual Delayed MAB (CD-MAB) jointly optimizes adaptive modulation and transmission power based on both channel state feedback and its Age of Information (AoI), thereby maximizing throughput. At the slower outer level, a Feedback Scheduling MAB dynamically adjusts the channel-state feedback interval according to throughput dynamics: stable throughput allows longer update intervals, while throughput drops trigger more frequent updates. This adaptive mechanism reduces feedback overhead and enhances responsiveness to varying network conditions. The proposed bilevel framework is computationally efficient and well-suited to resource-constrained UWA networks. Simulation results using the DESERT Underwater Network Simulator demonstrate throughput gains of up to 20.61% and energy savings of up to 36.60% compared with Deep Reinforcement Learning (DRL) baselines reported in the existing literature.