First go into the hall on the left side of the town and enter the "Heroes"-Section. There you will find all the different characters you can unlock using the coins that sometimes drop when you defeat enemies. You'll find that in one of the rows, instead of pedestals you can place your coins on, there are a couple of pictures and an arcade machine.

Interacting with that slot machine prompts you with 10 questions you have to get right to unlock Rin. Since guessing and hoping isn't much fun here are those questions with their respective answers:

• In TBC 1, why did Rin Satsuki decide to set off for the incident?
  Answer: Reimu's being a slacker again.
  
  
• In TBC 1, what's Rin Satsuki's occupation?
  Answer: A nurse
  
  
• In TBC 1, after resolving the incident, where did Rin Satsuki receive her celebration party?
  Answer: In Scarlet Devil Mansion (Relic)
  
  
• In TBC 1, what is the cost for Rin Satsuki to defense herself using the Flower Sign?
  Answer: Firepower down
  
  
• What is Satsuki clan's main business?
  Answer: Hospital
  
  
• In TBC 1, Rin Satsuki is the exclusive character of which mode?
  Answer: Another Story Mode.
  
  
• In TBC 1, what is the condition for Rin Satsuki's range being high enough so her bullet can fly beyond the border of the screen?
  Answer: Reaches Level 3 at Firepower
  
  
• In TBC 1, what's Rin Satsuki's ability?
  Answer: Manipulation of wind and flower.
  
  
• In the prequel's alternate story of Rin Satsuki, why Reimu didn't participate in the incident's resolution?
  Answer: The protagonist resolved the incident alone, which is customary.
  
  
• What is Satsuki's personality?
  Answer: A short-tempered little sis

And with that, you have successfully unlocked Rin!