Looks amazing! I'm super jelly. Out of curiosity, are you planning to add some kind of warping or smearing to compensate for the few frames?
Thank you.
With the NES hardware in mind, I could only do a few things. (as far as I'm aware, so prepare your grains of salt.)
First, movement in general is handled by the game code, so positioning things on screen on a frame-by-frame basis and how smooth that transition between frames looks is basically a matter of having high(-ish) framerate and positioning the sprite(s) in small increments.
I could've tried to simulate that in Photoshop (and I might) but I was trying to be quick and "git it dun" so I could have an idea if those frames of animation were working, if I would have to change anything, adding more frames or what. I'm thinking those are plenty, though... Dunno how things are going to look like in the morning.

In terms of adding smears like what I did with the sword slash animation (which btw desperately needs a revision...), I think I can do that to some extent (obviously drawing those smears by hand), but then it could be hard to reuse bits of those sprites in other frames.
Let me try to define a few things here in order to try and make things less confusing (?!?)
This is Ninja Gaiden III running on the FCEUX emulator:

The smaller window is the PPU viewer, which shows all the tiles used on that screen. Each of them is 8x8 in size.
On the left, the background tiles, and on the right the sprite tiles. Underneath that some options that don't matter much to the discussion here, and at the bottom all the color palettes used in that frame. In the first row those reserved for the background tiles and in the second those for sprites.
Those sprite tiles are arranged in a way and stored in memory (somehow, dunno exactly how those things work) to form "meta tiles" which in this case are the frames of animation for the respective sprites.
Here are some I arranged inside of Photoshop:

There you can see how a lot of those are reused to make different frames and be efficient in order to use the least amount of space in a cartridge as possible.
Now, those are not all of the tiles that game uses for Ryu's animations (and there are no enemy sprites there either), so those graphics are swapped for a different "page", unlike Super Mario Bros, which (again, as far as I'm aware) has all of its graphics assets in two of those pages... And I don't know how many of those I will, eventually, have available for me...
So, yeah, I want to do the best I can while still being reasonable, trying to make a shippable game, but I don't know yet how much I could fit in a cartridge.
And the NES hardware is really limited by today's standards, I can only flip those sprite tiles horizontally or vertically, not rotate them, so anything that should indicate rotation would be limited by that and me drawing new graphics to accomodate it.
Thus why in the "flip" animation I had to make a 45° frame, otherwise it would be very choppy.
I've taken a look at the NES sprites for the original Prince of Persia after you mentioned it, and that uses a lot of tile pages for the Prince (not to be confused with The Artist Formerly Known As Prince.

)... but that may be a very special case.