Yes, "it doesn't work" is a bit sparse.
Good thing you provided the sketch. Also a description of how you connected the potentiometers (or pots). Which sounds right btw. But no description of what kind of LED(s) you have (as AWOL is suggesting, common cathode or perhaps common anode), or if you use any resistors connected to them. Which you really (really!) should, so you don't destroy your Arduino. One resistor pr. Arduino output, to the LEDs different color pins. Then the common LED pin to either GND or Vcc, depending on the type of led. (If common anode, to the Vcc, and then you should also invert the analogWrite values - which in that case would be to take 255 and subtract your remapped pot_color values)
Anyway, it seems you forgot to define the output pins as output:
Which is the only thing I noticed atm.
This could be lucky if you didn't use any resistors, as the Arduino have an internal "pull-up" resistor at around 20 k ohm that is turned on if an INPUT pin (as an undefined pin would be) gets written a "1". But if this was the case, you should (maybe) notice a very dim LED.
Btw, don't feel bad for not having the newest one, the main difference as I understand it, is that the new Arduino Uno have a built-in USB connector (so no need for an external USB board with an FTD chip or similar).