how to automate DB restore to another server?
The best suitable solution in your scenario is LOG SHIPPING which is available in Web, Standard and Enterprise editions. I assume that your SQL server is not express as you mentioned There is already a scheduled job

If you want work with T-SQL, i don't think appending backup files is good idea, so you may want to change schedule job to create separate file for each backup. Then you can start with following query to get dynamic restore command based on backup history:

SELECT      [b].[database_name] ,
            [b].[backup_start_date] ,
            [b].[backup_finish_date] ,
            f.physical_device_name,
            ((b.compressed_backup_size / 1024) / 1024) as CompressedSize_MB, 
            CASE    WHEN Type = 'L'
                        Then 'RESTORE LOG '+ QUOTENAME([database_name]) +' FROM DISK=N''' + f.physical_device_name + ''' WITH NORECOVERY, REPLACE, STATS;  '
                    WHEN Type = 'D'
                        Then 'RESTORE DATABASE '+ QUOTENAME([database_name]) +' FROM DISK=N''' + f.physical_device_name + ''' WITH NORECOVERY, REPLACE, STATS;  '
            END as Script,
            b.is_single_user,
            b.user_name
FROM    [msdb].[dbo].[backupset] AS [b]
            LEFT JOIN  msdb.dbo.backupmediafamily f ON b.media_set_id = f.media_set_id
where   b.type in ('D', 'L') 
        and b.database_name = 'TestDAG' --- add your database name here
order by b.backup_finish_date desc