If you have a letter from SAAS already you don't have to pay any tuition fees to the University. Everything is taken care of directly between the Uni and SAAS. As KezzaN said, just take the letter to the Old Union Reception when you arrive at St Andrews during Fresher's week, or send them a copy of your letter by email. Last year I never even had to give them the letter since I was told by the Uni that my tuition fees had already been paid by SAAS before I even got to St Andrews. You only have to pay the University when you forget to apply to SAAS in which case you must pay yourself in order to matriculate and if you apply later during the year they will usually pay the Uni and the Uni will give you your money back. This is what happened to me in my 1st year since I only found out about SAAS close to the end of the 1st semester.
In terms of payment dates... I don't know if they are supposed to show as due on the website, or if they only show after the date has passed. What I do know is that you must have paid both your tuition fees and accommodation fees, or setup some sort of payment plan with the Uni, in order to matriculate. Since you must matriculate by the end of freshers week I guess the limit date for payment would be around the end of that week (24th September if i'm not mistaken).