Can I run a server for someone else under my own account?

Yes. You can create a server on your account and give the other person full subuser access via the Subusers page. They get their own login to the panel and can manage the server as if it were their own, within the permissions you grant.

Billing stays on your account. If you want the other person to have full billing control, a server transfer (contact support) is the right approach instead.