There's no restriction on using the mug on your first night's stay. In fact, you don't even have to be staying at a Disney resort to buy, activate and use a mug. If you're getting a complimentary mug because you're on a dining plan, you can get it activated and use it on the first day of your reservation.
There is a chip in the mug that works with a timer in the soda machine, and makes you wait a couple minutes between refills. Other beverages like coffee, you can drink as fast and as often as you can pour it.