Drupal installation in nginx subdirectory
I had a similar scenario, but in my case I didn't need to define again the root folder for the Drupal installation as you do on location /admin. 

In my case I only needed to add the nginx directives for the Drupal path:

location /admin {
    try_files $uri $uri/ @rewrite;
}   


location @rewrite {
    rewrite ^ /admin/index.php;
}