Oil spill incidents pose severe threats to marine ecosystems and coastal environments, necessitating rapid detection and monitoring capabilities to mitigate environmental damage. In this paper, we demonstrate how artificial intelligence, despite the inherent high computational and memory requirements, can be efficiently integrated into marine pollution monitoring systems. More precisely, we propose a drone-based smart monitoring system leveraging a compressed deep learning U-Net architecture for oil spill detection and thickness estimation. Compared to the standard U-Net architecture, the number of convolution blocks and channels per block are modified. The new model is then trained on synthetic radar data to accurately predict thick oil slick thickness up to 10 mm. Results show that our optimized Tiny U-Net achieves superior performance with an Intersection over Union (IoU) metric of approximately 79%, while simultaneously reducing the model size by a factor of $\sim$269x compared to the state-of-the-art. This significant model compression enables efficient edge computing deployment on field-programmable gate array (FPGA) hardware integrated directly into the drone platform. Hardware implementation demonstrates near real-time thickness estimation capabilities with a run-time power consumption of approximately 2.2 watts. Our findings highlight the increasing potential of smart monitoring technologies and efficient edge computing for operational characterization in marine environments.