Yes you are able to make a payment on your account before the first bill statement. Just go in the Billing section to make a credit card payment, you are able to enter an amount you would like to put on your account.
I would personally wait until the first bill comes in just so you can get an idea of how much your bill comes up to every month before you put money on the account.
Yes, you never want to pay for something until you have to :) Best to leave your money sitting in your savings account earning you interest, and then pay your bill the day before the due date. Unless you are the type to spend any money you have if its available to you.