To unlock Time Trial and Survival Modes, first clear the game. After defeating the final boss, select the file that you've beaten the game on and highlight Stage 7 (Coastal Battle) on the world map. Pressing up or down on this stage will allow you to play these special modes.

Unlockable	How to unlock
Survival Mode	Clear the game, then press "up" from Stage 7 on the map.
Time Trial Mode	Clear the game, then press "down" from Stage 7 on the map.