You'll need to remove the header over your slide to access the motor, the pullies, and the end of the cable where you tension it. You may also need to remove a side cover (trim) from the side of the slide (inside) so you can actually see the pullies.
If the cable is still there, ideally, you clip the old cable on the outside and use that cloth elastic sleave with a bit of tape to attach the old cable to the new one. Then it's simply a matter of pulling the new one through by pulling the old one out from the inside.
However, if the old one is broken and wires frayed, it's quite a different matter. Fishing the new cable through everything can be a real challenge. In theory, it's REALLY is very easy. In actual practice, space is so tight, it makes the job a horrible experience.
There are many, many YouTube videos on how to replace slide cables (bunches of them). Take some time and watch them. Like you said, you've got time, it doesn't have to be done right this moment. So, take some time and watch the videos until you get a good understanding of, not only how to re-install a broken cable, but also how to adjust, tension, and correct the slide alignment.
I've fixed my own a few times now (yes,... a few times), and currently have one that's started to fray. I do have the trailer in the shop right now for repacking the bearings and since it was there, I'm having them replace that frayed cable. I can do it, but I just don't feel like being a contortionist again, since it's in the shop anyway.
One thing I started doing was fixing that header on each of my slides so they can be easily removed. Here's how I did mine:
Linear Mode
