The increasing energy consumption of global software systems has established sustainability as a core requirement in software engineering. Digital services currently account for approximately 4% of global carbon emissions, yet practical architectural guidance for reducing the environmental impact of Web APIs remains scarce. This work evaluates four green design patterns - Green by Default, Wish List, Wish Template and Just Latest Updates - applied to the OpenDataHub API, a large-scale open data platform, as a case study. Using payload analysis and the Software Carbon Intensity metric, we measure the impact of each pattern on API energy efficiency and carbon footprint. Our evaluation reveals that the effectiveness of green software patterns is highly dependent on the granularity of data exchange. We conclude that software sustainability cannot be achieved through server-side optimization alone; instead, it requires an adaptive architectural approach that aligns data delivery with specific client needs to prevent over-fetching and reduce the overall carbon footprint of web services. Based on our findings, we propose a generalizable catalog of green design patterns for REST and OData APIs.