Learn English – n idiom about “dying monkey”

I can only speculate, but this simple hyperbole was probably generated many times for the same reasons:

• I would die to meet you. => I am dying to meet you.

as Jez mentions similar transformation happened in 'dying for...'

• I would die to have some strawberries. => I am dying for some strawberries.

Dying to is listed in CDO as

To be dying to do something, or for something, is to be eager to do or to have it.

Etymology might be related to several paths:

• dying wish - considered a wish that should be granted, so construct such as: "I want to meet you and I want it so much as if it was a dying wish. I am dying to see you!", though far fetched might actually be related
• suffering because you lack something (food, love, etc..); for example you could actually and literally be dying for some food, so such use is obviously justified. From this a hyperbole can be applied to any object of desire or need.
• I believe that it is quite natural to express strong desire by comparing it with one's own well being; to the question of "How much do you want it/that?", there is a very natural answer that means completely and that is to say "I would give my life to have it."

One for the road is the usual phrase.

It isn't just used in a pub, it is used when offering a visitor a final drink before leaving.

It is becoming less commonly used as drinking and driving is becoming less common.