Is there a way to...

[Tuzz]

Find out what the top ten singles in Japan were during the month of September last year?

Thanks in advance!

[Mulboyne]

The Oricon chart is the benchmark but they don't seem to archive their records in a way to answer your question (for free at least).

You can see the 4 weekly charts for that month on the Karao chart (the link starts with the ranking to the 4th September) which will give you a fair idea of the titles. There are no hard sales figures, though. The TBS CDTV chart archive is set up the same way so you can easily cross reference.