in order to play phantasy star 2 (among other games) multiplayer over wifi, you have to purchase an online entitlement from the playstation store.
all PSP games that have online multiplayer require this ever since socom FTB3. there's no way around this.
You can't play Phantasy Star Portable 2 with Xlink Kai.
Also like Shabutie said, you only can go online if you bought the game or bought the Little Wing License. Back when PSP2 was released, you could go online due to the servers messing up but that's all fixed now.