Honestly I just feel that it has to be something to do with linking with Alaya (The World) for mana and power.
Before you completely bash my idea and although it may start to sound far-fetched but the reason why I think so is because he is not the only one who hair colour and skin colour has changed.
An alternative version of Kiritsugu Emiya also has white hair and tanned skin and is also a counter guardian. Moreover, Okita Souji alter (counter guardian) also has tanned skin (I am not going to include her white hair because the original one already seems to have the same hair colour)
So maybe is because of being a counter guardian??? But of this is correct I am not very sure how to explain why some counter guardians like Asagami Fujino does not habe white hair.
What do you all think?