Note: This game is also titled Sengoku Musou 2 With Moushouden.
The quest for Kojiro's fourth weapon is in his fourth stage in Story mode. In order to unlock his weapon, you must clear 300+ kills as well as kill all Vagabonds (including any that spawn), help the Ieyasu retreat, kill Hideyori, and reach the southeastern gate within ten minutes. To do this, you will need a fast horse and a decently leveled Kojiro. At the start of the battle, clear all the Vagabonds around Ieyasu. Then, run back and clear the escape point. Wait for two more Vagabonds to appear. Kill one, which will cause at least six more to appear. Kill them, and wait for Hideyori to spawn in the southeastern maze-like area. Find and kill him, then quickly go to the southeastern gate. Musashi will appear in a short intermission sequence. A notice will then appear, showing you the location of your new weapon. Make sure to get your new weapon before killing Musashi.
At the main menu, select the "Vault Menu" option. Then, go to the "Weapon Acquisitions" menu. There are hints on obtaining the fourth and fifth weapons, but only for Gracia, Katsuiee Shibata, Kojiro Sasaki, Motochika Chokosokabe, Toshiee Maeda, and Yoshimoto Imagawa. To obtain other characters hints, import the Samurai Warriors 2 disc.