Existing music captioning methods are limited to generating concise global descriptions of short music clips, which fail to capture fine-grained musical characteristics and time-aware musical changes. To address these limitations, we propose FUTGA, a model equipped with fined-grained music understanding capabilities through learning from generative augmentation with temporal compositions. We leverage existing music caption datasets and large language models (LLMs) to synthesize fine-grained music captions with structural descriptions and time boundaries for full-length songs. Augmented by the proposed synthetic dataset, FUTGA is enabled to identify the music's temporal changes at key transition points and their musical functions, as well as generate detailed descriptions for each music segment. We further introduce a full-length music caption dataset generated by FUTGA, as the augmentation of the MusicCaps and the Song Describer datasets. We evaluate the automatically generated captions on several downstream tasks, including music generation and retrieval. The experiments demonstrate the quality of the generated captions and the better performance in various downstream tasks achieved by the proposed music captioning approach. Our code and datasets can be found in \href{https://huggingface.co/JoshuaW1997/FUTGA}{\textcolor{blue}{https://huggingface.co/JoshuaW1997/FUTGA}}.