How do you unlock rouge the bat in sonic advance 3?

You can't unlock Rouge the Bat in Sonic Advance 3. She is not a playable character in that game.

The playable characters in Sonic Advance 3 are:

* Sonic

* Tails

* Knuckles

* Amy

* Cream

* Chris (Only playable in the special "Chris Mode")

Rouge appears as a boss character in the game, but she's not playable.