Win 10 or more battles in Survival Mode to unlock EX Survival Mode. It can be done with any character, and whenever you want.
Hanatarou
Use Ichigo or Bankai Ichigo.
Iba Tetsuzaemon
Use Rangiku.
Ikkaku
Use Renji or Bankai Renji.
Isane
Use Hinamori.
Isshin
Use Chad.
Karakura Heroes Mach
To unlock this rather special mode, simply beat all 20 missions (under Mission Battle > Story Mission)
Karin
Use Chad.
Keigo
Use Ichigo or Bankai Ichigo.
Kojima Mizuiro
Use Ichigo or Bankai Ichigo.
Kon
Use Rukia or Shinigami Rukia.
Kuukaku
Use Yoruichi.
Tousen
Use Gin.
Unlock Extra Partners
Play Soul Road (Hard or Normal) and answer the question correctly, which the answer will be the name of the character you will unlock. The question will only appear when you hit on the blue spot with at least 60% Trust or more.