It’s like pushing a rope 🙂
Best idea is either to use the old cable as a leader and tie the new one to it, but I’m guessing it’s too late for that 😕
Alternatively take the grommet things out and thread through from the front (pivot end of the swingarm) and aim for the back. I think there’s internal baffles at the front and also as the swingarm tapers it sort of guides the cable towards the exit hole for you. You should eventually be able to catch the end of the cable with a bradawl or old spoke or something like that and guide it through.
I’ve also heard of people using loudspeaker magnets as the gear outer is magnetic and swingarm isn’t but didn’t quite have to resort to that.