Acne on back and Arms is commonly seen in people with dandruff or ones who go for physical removal of unwanted hair such as waxing or threading. As you rightly observed some times animal fur or if the animal is having dandruff also can trigger Acne.
Next Steps
1)If you are having dandruff, see the nearby skin specialist who will be able to treat your Acne and dandruff together.
2)If you have been waxing or threading, please stop that and go for Laser Hair removal treatment.
3) Avoid junk food , eat more of fruits and vegetables.
4) If you are petting your cat, make sure that you wear full sleeved dress during that time.