Welcome to the forums!
A picture of your budgie would definitely be helpful, maybe you can take one once your little newcomer is more settled in.
If the cere has a light bluish colour mingled with white and white circles around the nostril area, then you have a budgie girl.
Do you mean your cat got to the cage and managed to rip the tail feathers out of your budgie?
For your budgie's safety it really is best to place him/her on a room where your cat can't have access to. Even when your budgie is in the safety of the cage.
If your budgie is bleeding, you can stop it by applying some corn starch on the affected area.
If there is bruising or visible wounds from the cat attack, you should take your budgie to an avian vet in order to avoid serious life threatening consequences.
Best of luck with everything! I hope your budgie will feel better soon from the ordeal with the cat.