Brain development in the first few months of human life is a critical phase characterized by rapid structural growth and functional organization. Accurately predicting developmental outcomes during this time is crucial for identifying delays and enabling timely interventions. This study introduces the SwiFT (Swin 4D fMRI Transformer) model, designed to predict Bayley-III composite scores using neonatal fMRI from the Developing Human Connectome Project (dHCP). To enhance predictive accuracy, we apply dimensionality reduction via group independent component analysis (ICA) and pretrain SwiFT on large adult fMRI datasets to address the challenges of limited neonatal data. Our analysis shows that SwiFT significantly outperforms baseline models in predicting cognitive, motor, and language outcomes, leveraging both single-label and multi-label prediction strategies. The model's attention-based architecture processes spatiotemporal data end-to-end, delivering superior predictive performance. Additionally, we use Integrated Gradients with Smoothgrad sQuare (IG-SQ) to interpret predictions, identifying neural spatial representations linked to early cognitive and behavioral development. These findings underscore the potential of Transformer models to advance neurodevelopmental research and clinical practice.