Can Botox injections in my face cause pain in my arm?

The Botox injection should not have caused your arm pain. They are unrelated. You need to follow up with your primary care Dr  for a evaluation. 

I cannot think of any way that Botox injections in the face could cause pain in the arm.  I would recommend searching for other causes of the arm pain.  

Botox is not causing you arm pain.  It does not travel and cannot be the reason for your condition. I hope you get better soon.