Internet of underwater things (IoUT) is increasingly gathering attention with the aim of monitoring sea life and deep ocean environment, underwater surveillance as well as maintenance of underwater installments. However, conventional IoUT devices, reliant on battery power, face limitations in lifespan and pose environmental hazards upon disposal. This paper introduces a sustainable approach for simultaneous information uplink from the IoUT devices and acoustic energy transfer (AET) to the devices via an autonomous underwater vehicle (AUV), potentially enabling them to operate indefinitely. To tackle the time-sensitivity, we adopt age of information (AoI), and Jain's fairness index. We develop two deep-reinforcement learning (DRL) algorithms, offering a high-complexity, high-performance frequency division duplex (FDD) solution and a low-complexity, medium-performance time division duplex (TDD) approach. The results elucidate that the proposed FDD and TDD solutions significantly reduce the average AoI and boost the harvested energy as well as data collection fairness compared to baseline approaches.