Methane (CH4) is a potent greenhouse gas that plays a critical role in global warming, yet its relationship with climate variables remains underexplored, particularly in Eastern Saudi Arabia, which hosts over 70% of the nation's oil fields. This study presents the comprehensive analysis of the spatiotemporal variability of CH4 and its climatic drivers using TROPOMI/Sentinel-5P data from 2019 to 2024. Results reveal significant annual and seasonal fluctuations, with CH4 concentrations rising from 1892 to 1927 ppb. Seasonal patterns show peak concentrations in summer and autumn and lower values in winter and spring. Spatial trend analysis indicates an average increase of 5.09 ppb yr(-1) (MK-test). These variations are influenced by both anthropogenic and climatic factors. To examine the relationship between CH4 and climate variables, a geographical model is applied, achieving an overall R-2 of 0.82. We found that CH4 is negatively correlated with temperature, solar radiation, and wind speed, but positively correlated with precipitation and humidity. Specifically, the mean weighted regression coefficient between CH4 and temperature was -3.709, suggesting a decline of 3.709 ppb yr(-1) for each unit increase in temperature, which is likely due to enhanced oxidative processes under the region's high temperatures, driven by increased ozone and hydroxyl radicals (OH) that promote CH4 degradation. The random forest model further quantifies these relationships, yielding an overall R-2 of 0.79 and an root mean squared error of 2.934 ppb. The feature importance analysis identifies temperature and humidity as the most influential factors, followed by solar radiation, precipitation, and wind speed. These findings provide critical insights into CH4 dynamics in arid regions, offering valuable guidance for future research and policymaking.