The accurate prediction of bus arrival times at intersections is critical for intelligent transportation systems and urban traffic management. However, this remains challenging due to the complex interplay of spatio-temporal dynamics and multi-source uncertainties. To address this, we propose a two-stage prediction framework that integrates trajectory reconstruction with spatio-temporal dependency modeling. Structured sequences are built through trajectory interpolation, route projection, and intersection matching, decoupling the prediction into two sub-tasks: segment travel time prediction between intersections and delay time prediction within intersection areas. For the former, we develop the Spatio-Temporal Neural Network for Bus Travel Time Prediction between Intersections (STBTTI), which employs a Convolutional Long Short-Term Memory (ConvLSTM) encoder-decoder with self-attention. For the latter, we design the Spatio-Temporal Neural Network for Bus Delay Time Prediction at Intersections (STBDTI), incorporating feature enhancement, adaptive graph convolution, and multi-level attention. Experiments on real-world data demonstrate the framework’s superior accuracy and stability in predicting intersection-level bus arrivals.