How can I extract palm sugar from a container?

You could use the corkscrew in its classic role. Bore a hole, and then use the corkscrew to pull the sugar out in one piece. You should probably lay the tub on its side for this, as the sugar weight is probably too much to stay on the corkscrew.

The other option, though rather extreme, is to cut away the plastic tub with wire cutters or shears. That way, you're peeling the tub away, and will be left with the same lump, but no tub.

It sounds like gouging and digging are your main options. Referring to my own kitchen, my corkscrew wouldn't work, but the round honing edge I have would, my kitchen shears, a small chefs knife could, cutting downward, line up cubes to pry/yank out with something else rigid.

After that, I think @Carmi isn't too extreme with suggesting to cut the tub away; just break it up once done and put in a different container.

To remove palm sugar from the grips of its plastic container, remove the lid and place the container in a sturdy plastic bag (I use my food saver bags). Strike the sides and bottom of the container with a mallet. This will shake the cake loose. The bag serves to catch all of the sugar bits. You can also use it to store the cake.

I find scraping the sugar with a dinner knife works. Rotating it at a angle in a circular motion. So you end up with shavings.

place the jar in warm or hot water for a few seconds. The sugar will loosen up from the sides of the container. At that point it should b fairly simple to remove. Worked very well for me.

Try microwaving it briefly. That works with hardened brown sugar anyway. the goal is to warm it, not cook it or melt it though! Once it's softened consider re-portioning it into smaller containers/ or creating spoonfuls of the size you normally use and letting those cool, then storing those.