Here is a list i made of the weaknesses of each boss including willy's castle.
For willy's castle i only list the stage and the weapon required for the boss.
Don't forget to abuse of the save feature everytime you get to a boss room or during the stage when you get to hard parts.
1st boss that should be killed is Bombman.
boss/weakness weapon: (they are in the order you should fight them)
Bombman - Fireman(F)
Gutsman - Bombman(B)
Cutman - Gutsman(G)
Elecman - Cutman(C) (there is a gadget to collect during this stage,requires weapon G to reach)
Iceman - Elecman(E)
Fireman - Iceman(I)
Willy's castle:
1st stage: elecman (E)
2nd stage: Cutman: mega buster
Elecman: Cutman(C)
Megaman doppelganger: Fireman(F)
3rd stage: Gutsman(G) and mega buster
4th stage: boss rush
Willy's 1st form: Fireman(F)
Willy's 2nd form: Cutman(C)