Sql-server – Steal ownership on database from user that no longer exists

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.

Indeed what I was looking for (but were unaware of its name) is called cross database ownership chaining.

By reading the article Understanding Cross Database Ownership Chaining in SQL Server I was able to solve my problem.

As a reference, below is what I did:

Use [DatabaseA]
GO

ALTER DATABASE [DatabaseA]
SET DB_CHAINING ON
GO

Use [DatabaseB]
GO
ALTER DATABASE [DatabaseB]
SET DB_CHAINING ON
GO

EXEC sp_grantdbaccess 'UserB';
GO 

GRANT SELECT, UPDATE on [DatabaseB].[dbo].[TableA] TO [UserB]
GO

Thank you all.