Thank you for saving a life! ♥ I love you for that!
It depends on if they are doing the spay there or giving you a certificate to get her spayed. At the shelter where I volunteer, we do the spays prior to the dog going home. Do you know when her spay is scheduled? They will likely keep her overnight after the spay and then you can pick her up with instructions on how to care for a dog after surgery and things to look for in case there is a problem with the incision. (rare)
When you go tomorrow to pay, ask them then. It is in my experience that most shelters have their own scheduling. (some only do neuters on Tuesdays, etc.)
Have you already been approved? (home visit, references checked, etc) If not, that could add a couple of days.
Best wishes to you!