Sanzu no Kawa, is literally translated from Japanese into English as the "River of Three Crossings", which, for people in English, it's known as the River 'Styx', shortly put, Solstice is cryptically saying that they're all upon the pathway to Hell and will be going to it one way or another. This is meant to show how dark Solstice is compared to the usual Solar people see. Rose only references Styx as just "Styx" herself, therefore she may or may not truly understand what Solar's referring to. Mainly since she's used to the English sayings of Hell and Styx, where in Japanese; would be respectively, Makai, and Sanzu no Kawa.
Please do note that no one could truly understand Solstice upon speaking Japanese, as Rose's understanding of the language is fairly limited, and there is no other person that seems to be able to translate with him. Much like how Solstice has limited understanding of French, German as well, though can seemingly understand Japanese a bit more, and let alone, able to speak sentences of it. Though seeing how Rose is a Princess of Hell herself, she could actually know what he's saying regardless due to Hell being, ironically, like America, a huge melting pot of culture.
Please note that for any character other than Rose, seeing they can't understand Japanese, upon users reading this information, shall not transfer their information right into the characters they're Roleplaying as. This will count as Metagaming.