How do I change the admin for family sharing when admin lost their phone

An Apple ID password can be reset, even on a borrowed device, using the Apple Support App, check this article:

If you forgot your Apple ID password - Apple Support

Note that it will be a lot easier if he uses the method proposed by Ingo2711. Changing the Family Organizer involves closing the entire family group and starting it again, with added complications to each user if you are using subsciption services, and special steps required if minor children are involved.