To unlock Extra Mode in Mountain of Faith, you need to clear the game on Normal difficulty or harder without using continues. Note that this is pretty much the standard procedure for unlocking Extra in any applicable Touhou game.
It's easier said than done, I assure you. It will take practice, it will take patience, and it will take time. But you'll get it eventually. Once you do it on one game, it gets a little easier to do on other games because your basic skills improve. Memorization of the danmaku and understanding the game mechanics is vital.
Good luck. Say "hi" to Suwato for me.