If I want to start a 1-BR apartment lease on September, when should I start looking for it? And is it normal to start the lease on September?
I am planning to bring my family and already prepared everything including the VISA and the flights, but I haven’t found the 1-BR or Studio apartment for us.
If you have any recommended place or know somebody that will pass-down his lease, please let me know. Thanks!
I would certainly start looking sometime in the summer, at least a month or two in advance if not more. There is nothing abnormal about whenever you start your lease, it always depends on when the place is available and what your requirements are. When I first moved to Chicago, my lease started on Sept 1, while in my current apartment it has a Sept 21 start date.
About the rest of your question, I hope you are aware of:
This is where most people from the university go looking for places, but there are certainly other options too, including word of mouth that you are already utilizing. 
Another practical suggestion I have if you feel more comfortable with a particular management company or a high-rise management is to make an appointment with them and share your requirements with them. As they know when the leases of their tenants are ending, they can share options that fit your time and space requirements very easily.
Thanks Kartik!! 
So far I found one from uChicago fb group (“free and sale”