In Oracle, how can I give one user privileges to alter the schema of another user

I have dealt with this issue before.

When you ran

select count(1) UserTableColumnCount from information_schema.columns
where table_schema='mysql' and table_name='user';

you should have gotten 42. That's how many columns MySQL 5.5 has for mysql.user. Since you got 39, that means you must have upgraded from MySQL 5.1. That has 39 columns.

I wrote an earlier post about the number of columns in mysql.user in different versions : MySQL service stops after trying to grant privileges to a user

Here is the post where I dealt with this : mysql: Restore All privileges to admin user

Hopefully you could run

# mysql_upgrade --upgrade-system-tables

to realign mysql.user and have it autofill missing permissions with Y.

Give it a Try !!!

If you want to try to fix the mysql.user manually, here are the steps:

#
# Backup the mysql.user table
#
CREATE TABLE mysql.user_backup LIKE mysql.user;
INSERT INTO mysql.user_backup SELECT * FROM mysql.user;
#
# Add Missing Columns
#
ALTER TABLE mysql.user 
    ADD COLUMN Create_tablespace_priv enum('N','Y') NOT NULL DEFAULT 'N'
    AFTER Trigger_Priv
;
ALTER TABLE mysql.user ADD COLUMN plugin char(64);
ALTER TABLE mysql.user ADD COLUMN authentication_string text DEFAULT NULL;
#
# Give root user all privileges
#
UPDATE mysql.user SET
Select_priv='Y',Insert_priv='Y',Update_priv='Y',Delete_priv='Y',
Create_priv='Y',Drop_priv='Y',Reload_priv='Y',Shutdown_priv='Y',
Process_priv='Y',File_priv='Y',Grant_priv='Y',References_priv='Y',
Index_priv='Y',Alter_priv='Y',Show_db_priv='Y',Super_priv='Y',
Create_tmp_table_priv='Y',Lock_tables_priv='Y',Execute_priv='Y',
Repl_slave_priv='Y',Repl_client_priv='Y',Create_view_priv='Y',
Show_view_priv='Y',Create_routine_priv='Y',Alter_routine_priv='Y',
Create_user_priv='Y',Event_priv='Y',Trigger_priv='Y',
Create_tablespace_priv='Y'
WHERE user='root';
FLUSH PRIVILEGES;

That's it !!!

You need to grant EXECUTE on package itself. If security is your main concern you may create a new package that has just types, but you cannot specify that a given user can access only some objects defined in package specification - either all or none.