I graduated BLC on 7 Aug but I know cut off is next day. Will my 1059 get processed on time for Sept Promotion or will it be in Oct?

No. Don't worry about when you will get promoted. It will come. It may be two or Three months before you receive promotion orders. That is of no importance. What is important is your date of promotion, often backdated a moth or two. Makes for a nice one time chunk of change. Be sure your records reflect your "date of rank".

If you got your S1 to update it by yesterday, it's possible. Otherwise, October.

No. You will have to wait for it to hit until the next month. Nothing goes through the system that fast.