Move Set
I will describe possible moves for Ryu Hayabusa for most button combinations. I'll use GIFs I've made from videos taken from playing Ninja Gaiden 2 NES, (Ninja Gaiden 3 for the side A upgraded sword move and Double Dragon 3 for a couple of jump attacks) on an emulator. So you can have a visual representation of the moves. This I feel is a better way to understand the character.
A - Dragon Sword
Down + A - I have not thought of a move for this. Possibly just a ducking sword stab.
Left + A or Right + A - Dragon Sword Upgraded
Smash + A - Hard Kick
Up + A - Spin Slash (also works as a recovery move)
Left + A or Right + A while in the air - Aerial Dragon Slash or an Aerial punch/kick (I have GIFs for both options)
Aerial Recovery Move - Spin Slash. This means Ryu has no 2nd jump. Just one jump and the recovery. This is because I feel a 2nd jump with the wall climbing is a little OP.
B- Shuriken
Left + B or Right + B - Windmill Shuriken
Hold B for a few seconds without moving - Create a clone. The clone lasts for a short time (different to NG2) and only does 50% damage but mimics all of Ryu's moves.
Up + B - Fire Ninpo
Down + B - Fireball Ninpo
Throw - Nothing fancy here, just an average throw.
Block - Ryu parries the enemies moves with his sword
Super special ability from Smash Ball - Art of the Fire Wheel Ninpo - Makes Ryu invincible for a limited time and damaging anyone who comes into contact with the Fire Wheel. You could also majorly slow down the sun speed of every other character to make it easier for Ryu to run into everyone else and cause major damage.
Character Ability - Wall Climbing