Precipitation nowcasting (short-term forecasting) is still often performed using numerical solvers for physical equations, which are computationally expensive and make limited use of the large volumes of available weather data. Deep learning models have shown strong potential for precipitation nowcasting, offering both accuracy and computational efficiency. Among these models, convolutional neural networks (CNNs) are particularly effective for image-to-image prediction tasks. The SmaAt-UNet is a lightweight CNN based architecture that has demonstrated strong performance for precipitation nowcasting. This paper introduces the Multimodal Advection-Guided Small Attention GNet (MAD-SmaAt-GNet), which extends the core SmaAt-UNet by (i) incorporating an additional encoder to learn from multiple weather variables and (ii) integrating a physics-based advection component to ensure physically consistent predictions. We show that each extension individually improves rainfall forecasts and that their combination yields further gains. MAD-SmaAt-GNet reduces the mean squared error (MSE) by 8.9% compared with the baseline SmaAt-UNet for four-step precipitation forecasting up to four hours ahead. Additionally, experiments indicate that multimodal inputs are particularly beneficial for short lead times, while the advection-based component enhances performance across both short and long forecasting horizons.