Maintaining or Re-Gaining access after SQL Server database restore without SYSADMIN rights?
to relink the login to the correct id you will have to run (I assume that the login is appuser, and the databas user appuser):


  alter user appuser with login = appuser ;


From BOL:

To change the name of a user requires ALTER ANY USER on the database. To change the default schema requires ALTER permission on the user. Users can change only their own default schema.
Requires CONTROL permission on the database to remap a user to a login

As you explained that you have the db_owner role on the database, it should work.