If you draw on or draw upon something such as your skill or experience, you make use of it in order to do something. He drew on his experience as a yachtsman to make a documentary programme
gain inspiration from, take ideas from; take advantage of (resources), make use of (support or help); suck in; approach, come near to; make marks on a surface with a writing implement
As a period of time draws on, it passes and the end of it gets closer. As the afternoon drew on we were joined by more of the regulars. = wear on