The name Old School was a reference to the term of the same name, which is slang for "retro" or "old". In this context, Old School might be a reference to the original Plants vs. Zombies, where plants could not be revived.
The best way to earn Old School is to replay level 1 of The Boonies. Here, you must keep your plants from being stunned by the zombies. The best plant to use is an Aspearagus because it can kill regular zombies in three hits and from a safe distance. Freeze the last zombie before it gets to your Aspearagus to win.