Accurate assessment of atmospheric nitrogen dioxide (NO$_2$) and sulfur dioxide (SO$_2$) is essential for understanding climate-air quality interactions, supporting environmental policy, and protecting public health. Traditional monitoring approaches face limitations: satellite observations provide broad spatial coverage but suffer from data gaps, while ground-based sensors offer high temporal resolution but limited spatial extent. To address these challenges, we propose PollutionNet, a Vision Transformer-based framework that integrates Sentinel-5P TROPOMI vertical column density (VCD) data with ground-level observations. By leveraging self-attention mechanisms, PollutionNet captures complex spatiotemporal dependencies that are often missed by conventional CNN and RNN models. Applied to Ireland (2020-2021), our case study demonstrates that PollutionNet achieves state-of-the-art performance (RMSE: 6.89 $μ$g/m$^3$ for NO$_2$, 4.49 $μ$g/m$^3$ for SO$_2$), reducing prediction errors by up to 14% compared to baseline models. Beyond accuracy gains, PollutionNet provides a scalable and data-efficient tool for applied climatology, enabling robust pollution assessments in regions with sparse monitoring networks. These results highlight the potential of advanced machine learning approaches to enhance climate-related air quality research, inform environmental management, and support sustainable policy decisions.