But if it is the first time of H1-B application for me, do I have to wait until the start date of Oct.1 even I got H1-B approved before Oct.1?
-No, non-profit H1 is not limited on the work start date. Your employer needs to indicate the work start date on the h1 application form (I-129) so that you can start to work from that day or work immmediately after the h1 approval if the approval is later than the start date indicated in the Form I-129.