Maybe not a simple conversion, because the PIC used there has two analog comparators, the Nano only has one. So either a different method needs to be used or an external comparator is required.
To make this transfer you have to count on at least 10-20 hours of engineering time. The price of that PIC programmer, plus the anticipated cost of the PCB and PIC itself, don't add up to pay for even the first half hour of engineering time.
That engineering time includes a few hours just for figuring out what the original program does and understanding how it's done. Then it has to be translated to an Arduino, the circuit itself has to be translated to the Arduino, and finally it requires testing. The last may be the hardest as you'd need the surrounding hardware as well.
That is, if indeed a direct translation is possible at all.
How much are you able/prepared to do yourself and do you know the requirements of the Korg Synthesizer ?
While it is true that the Nano does not have an equivalent for the PIC dual comparator feature, you can instead use 2 analog input pins to detect the falling edge of pulses measured at the LED pads.
Thus you can measure the frequency, generate a sync pulse output (double the input frequency) and align it with the input pulses.
From what I can see, you could even do it with an ATTINY85 but for development the Nano is easier to use.
Obviously it would be easier to use the PIC solution. But, if you want to start learning about Arduino, maybe this would be a good opportunity.