In Survival Mode (3rd option on title screen), defeat 50 enemies with a custom PF (the one you made in Story Mode).
Apollon
Complete Story Mode on either Easy or Normal.
Conversion-exclusive items
If you convert a save made from the trial version of J-Phoenix (4th option on the Story Mode menu), you gain the following exclusive items: AlsareaGS parts, Zanbatou (big sword) and Force Sword.
Level Select and Omake episode
Finish Story Mode once to unlock both. Choose MISSION to select your stage. Instead of getting money from e-mails, you get money right after completing missions. The Omake episode appears after episode 20 in the level select screen.
Master Difficulty
Complete all Normal Mode missions, including the Omake episode. On the level select screen, push R2 or L2 to change difficulties.
Politarn 3
After unlocking Alice, enter Survival Mode again and defeat 100 enemies with a custom PF.
Special Mode and special parts
Convert your save from the trial version of J-Phoenix, and use this save to finish the story mode in either Easy or Normal difficulty. You need to clear all stages, even the Omake. Now, on the level select screen, push L2/R2 to access the Special stages. They're set on the same maps as the normal story mode, but enemy placement is different. Also, there are specific conditions to get a special part in every stage. Those conditions are noted on the level select screen. The first one is usually "clear mission", and the second one is "having XXXX ammount of money". If you complete the conditions, you get a special part, not available anywhere else. Note: if you don't convert your save game from the trial version, there is no other way to unlock this.