Sequential point-of-interest (POI) recommendation aims to recommend the next POI for users in accordance with their historical check-in information. However, few attempts treat timestamps of check-ins as a core factor for sequence models, leading to insufficient insight into user behavior and subsequently suboptimal recommendations. To address these limitations, we propose to assign equal importance to both POIs and their timestamps, shifting the point of view to recommend the next POI and predict the corresponding timestamp. Along these lines, we present the Time-Aware POI Recommender with Timestamp Prediction (TAPT), a multi-task learning framework for explainable POI recommendations. Specifically, we begin by decoupling timestamps into multi-dimensional vectors and propose a timestamp encoding module to explicitly encode these vectors. Additionally, we design a specialized timestamp prediction module built on the traditional sequence-based POI recommender backbone, effectively learning the strong correlation between POIs and their corresponding timestamps through these two modules. We evaluated the proposed model with three real-world LBSN datasets and demonstrated that TAPT achieves comparable or superior performance in POI recommendation compared to the baseline backbone. Besides, TAPT can not only recommend the next POI, but predict the corresponding timestamp in the future.