Question

Do I need to activate Cloud Storage and turn on billing if I was granted access to someone else's bucket?

Answer 1

No, in this case another individual has already set up a Google Cloud project and either added you as a project team member or granted you permission to their buckets and objects. 

Once you authenticate, typically with your Google account, you can read or write data according to the access that you were granted.

For information about adding project team members, see Adding a member to a project.