Cheats
No Helmet Mega Man
During the Stage Select screen hover over Ryu's icon and hold the Shoot button then press Right, Right, Right, Left and Mega Man's icon should be helmet-less.
Change the Music
In any stage enter the the pause menu and hold the Jump button and press Up, Down, Down, Down and the music should change to "Goes With Everyting" which is Guile's theme from Street Fighter II.
Hadoken and Seiei Enbu
During the title screen hold the Shoot and Jump buttons for 6 seconds and a chime should occur.
Hadoken
Mega Man is able to shoot a hadoken just as Ryu would in Street Fighter by pressing 
+ 
(or quarter circle and the shoot button).
Seiei Enbu
While in a Stage press the Shoot button with the WeaponSwap1 button at the same time and you should hear a chime confirming that you did it correctly. Transparent Mega Mans now follow the real Mega Man and do as he does but only the last one is able to shoot.
Secret Boss
Get a Perfect on 3 boss fights and during the battle with M. Bison (Vega in Japan) Akuma will come out of the sky when M. Bison is almost defeated. Mega Man will now be teleported to Akuma's lair to fight.
Tip: To make this easier simply use an E-Tank to recover your life when you are about to defeat a Boss.