How can I refinish, update, or paint cedar ship lap walls?

You need to clean them but first you can use a deglosser to remove any sheen. Prime with a good primer such as Zinsser Bullseye 1-2-3 which can be tinted the color of your final coat and then apply your finish coat.

I personally wouldn't paint real cedar walls. Maybe they just need a good cleaning. If you are determined to paint them, sand, clean, prime then paint. Best of luck.

If you are painting it a light color, you definitely want to use a good primer so the color of the wood doesn't come through. I painted my pine paneling in my living room - https://www.hometalk.com/diy/decorate/walls/how-to-paint-wood-paneling-baseboard-trim-without-bleed-throughs-34306955?r=2

A small roller was super helpful to get it done quickly!

Sand and repaint.

Make sure it is clean of any oils then sand if it is rough and then vacuum or use tack cloth. Next use a bonding primer then paint. I recently painted a chair and had to use these steps if you want to see. However I am using milk paint and for your walls i would use regular latex paint. https://lifeonsummerhill.com/how-to-paint-with-milk-paint/

You should prime if you want to paint, but you can also glaze the wall with watered down paint or stain if you want to keep the style of having real wood in the room. Cedar has a lot of tannin that can make some finishes difficult so before you make your paint/stain purchases I suggest you talk to someone at a paint store. They will know what their products can do.