Yeah, J works. Assuming I'm not missing something, it would be nice to have the controls listed somewhere. Anyway...
This game definitely feels very authentic to the NES. Arguably a bit too authentic in some areas. The weird jumping does reflect old school games well but it can be annoying to get the hang of. I did grow to like it though - it makes navigating levels an interesting puzzle as you have to figure out where you can/can't jump. What I definitely didn't like was the lives system. It feels like it's there just to be retro and doesn't add anything other than annoyance. It's especially bad in this game since the levels are very specific and if you miss an enemy with a box you've basically wasted a life.
That gripe aside, this is a really well made game. The core mechanics are fairly simple but lead to lots of interesting challenges. I like how there are multiple ways to kill enemies (from the side, from above, into a pit) and how you can push enemies as well as boxes. The level design is all well thought out to take advantage of your abilities. There seems to be some clever details - I noticed that exiting an elevator with a box in front lets you collect the box for points. There's also an impressive amount of content though unfortunately I only reached level 13 out of 20.