user only have full control permission at document library, when user try to click on alert me setting then will show Error: Access Denied
this is by design on Microsoft product or maybe is bug :D
When you click on the Alert me you will route to the page sites/abc/_layouts/XXXXX , which is needed read permission at site level.
1. Grant user view permission at site level. > if allow user to see other information
2. If user using client Microsoft outlook 2007 then can direct sync the folder to outlook ( if have any new document upload then you will see unread document at outlook)
3. Admin can help user add the alert.