By the mid to late-1800s, the word "nut" was slang to mean a person's head. Not long after, it looks like it also acquired the meaning of someone who was not acting right in the head, e.g., a person acting strange might be described as "nuts" or being "off their nut."

This phrase refers to someone trying to escape something something by climbing up a wall. It is not known when it was first used. Seriously, no amount of Googling yielded any results. The phrase today means to be annoyed, irritated, be driven to irrationality, etc. Here are a couple of examples of things that drive ME up the wall...