Can you drop off rental car before office opens?

It depends on the company and the particular location. Some branches do not have a secure lot for this purpose. Airport rental car locations, however, often do since they are behind airport security. Contact the branch where you rented the car and ask what your options for this are.

Just be aware that, unless it is stated in writing, you are responsible for the car and any damage it sustains until it is inspected and signed off by the lenders agent, whether it is in secure storage or not.