I have a cake due Friday evening, and I plan to make and assemble it tomorrow. I have heard that royal icing breaks down from the fat in modeling chocolate, but how fast does this happen? Should I wait as long as possible before attaching the figures (The party is on Saturday) or should I do it earlier so the royal icing has more time to dry?

Not sure what kind of figures exactly you're making, but I've used coffee stirrers and toothpicks in the past to attach modelling chocolate to my fondant cakes.