I'm assuming that this visa was issued last year on Sept 15 or 16, 2008, therefore goes invalid on Sept 15, 2009. Then yes, you can enter on Sept 14, 2009. When you enter China, you will then be allowed whatever the duration of stay is listed on the visa (often 30, 60 or 90 days). Note that you do NOT have to be OUT of China by Sept 15.

You shouldn't have a problem with the airlines at check-in, but it's always possible to get an agent that's not up to snuff on the regulations. If it were me, I wouldn't bother getting a new visa for this. If you plan on making more trips to China in the next year and it suits your time frame, you can apply for a new visa, the Chinese Embassy/consulate will just cancel the current one when they issue the new.