I don't know the comics or movie, but I did download the Scott Pilgrim video game on PSN. I had heard it described as being heavily inspired by River City Ransom, and a modernized RCR was something I'd long wanted.
In many ways, Technos's River City Ransom for the NES remains the most evolved beat 'em up to this day. The game basically played like Double Dragon (also developed by Technos), but it also included many RPG elements. You could earn money by defeating foes, then go to shops to spend that coin on new moves, items, and stat boosts. The hip city setting was also slightly more realistic than in the average brawler, populated as it was by occasional friendly NPCs, who would offer hints to get you through the story. Instead of the typical series of straight-line stages, River City Ransom had a unified world that players could travel backwards and forwards through as needed.
Scott Pilgrim vs. The World is more linear than River City Ransom, but it has most of the rest of that stuff covered. Its debt to RCR is most immediately evident in the ability to pick up downed enemies or allies and use them as melee weapons. The controls are at least more sensible than Technos's two-button classic, but the feel is frustratingly nowhere near as sharp as the best Capcom arcade brawlers of the 90s, and the hit detection is punishingly specific. The game is also extremely buggy, and the very real glitches kind of take the humor out of the "Subspace" zones filled with flying piggy banks and fake graphical corruptions. A word of advice: if the music drops out suddenly, that's probably a sign that everything is about to go to hell, so you might want to just quit while you're ahead. Also, try very hard not to fall in any pits or throw enemies into them.
Like all beat 'em ups, the game is ideally played with friends, and Scott Pilgrim even allows for two players more than River City Ransom. Not only does it help to have other guys fighting on your side, but the game allows players a chance to revive downed comrades, thus saving them from having to lose precious lives. Even with two other players, however, I find that the aggressive enemies make it pretty hard to successfully resuscitate an ally before their countdown to death expires. In fact, the game probably throws larger groups of enemies at you than any actual vintage beat 'em up. Sometimes they'll even pile on top of you, and bursting them off you is one of the game's more memorable gags.
Even on the easiest difficulty, the game can be quite challenging. It does save your progress between stages, and you can revisit past levels to grind for money and experience, although it is annoying that the shops won't tell you what items do until after you've bought them.
Scott Pilgrim vs. The World's actual fighting is merely passable, but the music and art are impressive, and it is a new 2-D beat 'em up in 2010, which is nice to see. With more players, more characters, and a real save system, it may indeed be a better version of River City Ransom.