WELCOME TO ADULTHOOD! It all evens out.
Chances are the rent will not be the full amount, but rather a pro-rated amount to cover the last few days of the month since technically you paid until Sept 22. Then in October, you would pay the full amount again. Or it could be your lease still started on Sept 1. They just didn't care about you paying until you moved in but you would still owe all the month. Ask the office for clarification.
Student housing leases are often for the school year rather than a calendar year, and have a rent amount for that period (say, $5400 for 9 1/2 months) and then a payment schedule of installments paying off that amount.
What you don't make clear is your regular monthly rent amount. Is it $600?
I am 18 I stay in a student apartment I moved in on august 22nd which was the set move-in day we could not move in any earlier so rent was originally due on the first of august however they said it was ok as long as you pay it before you move in so I paid (600 damn dollars) about two days before move In day the rent is due once again on Sep. 1st why do we have to pay again when we just paid and we weren't even living here for a full 2 wks ???