In American English, the usual preposition to use is "on". Example: "He's willing to work on short notice." As Steve explained, "on short notice" means that you are willing to do something even though you are asked to do it a short amount of time before you have to do it. One more thing: Instead of the adjective "willing," the paragraph should use the noun "willingness".

"at" short notice; not "a". "Notice" is another noun meaning "notification". If I give you "notice" then I make you aware of something. In this case I'm making you aware that I need you in the office to do some work by X O'clock. "Short" means that X O'clock is a short time in the future. Hopefully that makes sense. :D