Effectively estimating the arrival day of a shipment is an important capability for an express courier company to ensure both customer satisfaction and internal operations efficiency. This paper studies predicting the estimated time of arrival (ETA) of a package, shortly denoted by PETAP, under a collaboration with an industrial partner. Our approach employs machine learning (ML) techniques: CatBoost, multi-layer perceptrons (MLPs) with categorical embeddings, and Transformer neural networks, to predict delivery dates based on shipment locations and status. Challenges such as complex inter-modal networks and high-cardinality categorical features are addressed. Our paper contributes to the literature by formalizing the PETAP problem in the context of express shipping: the proposed models outperform the current business baseline accuracy by more than 25%. In our experimentation with a dataset including millions of data-points we propose a tabular vs sequence to sequence approach observing the superiority of the former. Future research directions include explicit modeling of transportation networks and exploring alternative ML approaches for improved prediction accuracy.