Extracting temporal information is critical to process health-related text. Temporal information extraction is a challenging task for language models because it requires processing both texts and numbers. Moreover, the fundamental challenge is how to obtain a large-scale training dataset. To address this, we propose a synthetic data generation algorithm. Also, we propose a novel multi-task temporal information extraction model and investigate whether multi-task learning can contribute to performance improvement by exploiting additional training signals with the existing training data. For experiments, we collected a custom dataset containing unstructured texts with temporal information of sleep-related activities. Experimental results show that utilising synthetic data can improve the performance when the augmentation factor is 3. The results also show that when multi-task learning is used with an appropriate amount of synthetic data, the performance can significantly improve from 82. to 88.6 and from 83.9 to 91.9 regarding micro-and macro-average exact match scores of normalised time prediction, respectively.