I'm not going to tell my friend I hate doing something that she loves to do together with me
Are you serious? Why wouldn't you tell your friend that you hate doing something?
My friend gets upset very easily
You should be more honest with your friend and let her know how you really feel. If your friend finds out that you are not being honest and true she may not trust you in the future.
She will just get over it if she ever finds out.
But she may be even madder that you took so long to tell her. I think you should just be honest and upfront with her about the thing you don't like doing.