How do I get a single IK bone to move up when the controller pushes against it? So I have a minecraft rig that uses one IK bone for the arm.

What I want is for the bone to move up when the controller moves up as well, and when it goes down, for the IK bone to not go below its limits.

As you can see, moving the controller up and down doesn't currently affect the IK bone.

What I want is something like this, but when the controller moves. Also, I want the arm to return to position if the controller is moved forwards or sideways and for the IK to continue to be parented to the torso bone.
I hope this is clear enough of what I want. I definitely know in my head how I want the bone to act, and I feel like it is possible inside of Blender, but I've been messing around with the bones for a while now and I still can't figure it out.
The IK goal is to manage a chain of bones and bend them from point A to point B with an orientation towards point C. Here you have just one bone, you should use a damped track constraint instead, which has the simple role of pointing a bone towards another and seem to be what you need.
Then, before the damping track, you can add a Child of with only Z translation enabled. Don't forget to hit the set inverse button.