The following weapons were used in the anime film Princess Mononoke:
Ishibiya Hand Cannon
Japanese Ishibiya (石火矢) hand cannons are manufactured and used by Lady Eboshi's men. It is commonly believed that the first encounter of the Japanese with firearms was with the arrival of the Portuguese in the XVI century, however, it appears that earlier and more primitive guns like the Ishibiya were also used prior to this. The Ishibiya appears to have originated from the medieval Chinese hand cannon known as Chong or Huochong.
A rather primitive hand-cannon musket without any kind of lock is briefly used by Lady Eboshi to shot Moro.
XIV-XV Century Breech-Loading Handgonne Arquebus
Lady Eboshi and her men use early shoulder mounted handgonne matchlock arquebuses typical for XIV-XV century Europe. It also uses breech loading. While created back in the 14th century, breech loaders was not mass-produced until the mid-19th, due to the inability to produce it affordably and reliably in large quantities.
Jigo's men uses fuse bombs against the wild boars. They appear to be hewn out of stone and filled with gunpowder.