From personal experience on wig making her updo look will need to be make as a solid style meaning you just can't take it out again once it's done.
Your going to need two seperate wigs for her and extra wefts for the updo normally.
Chances are she is wearing wiglet/hair piece rather then her own hair for that look attached to her natural hair.
The wigs you linked convert to $50. I honestly recommend buying look with Arda for base with and match wefts.
Dark Copper Red would be a good color match.