Having read what CR suggests I agree, you can indeed use a single resistor. Since each LED will have a dedicated output pin on the arduino the single resistor will act as the common current control device. By using controlled switch-on you ensure that only one LED output at a time is actually active. I'd surmise a frequency of around 25Hz shouldn't be noticeable. To overcome brightness variation concerns you could drive each output via PWM rather than simple ON-OFF and effectively control the brightness between different LED colours. Since the PWM frequency will be much higher than the multiplexing frequency there shouldn't be any problem combining both switching frequencies.