Beethoven almost never dated his compositions. This morning's prelude is an exception, dated December 25, 1821.
Much has been made of his ability to compose 'in his head' without having to 'pound it out' on a piano, but that is in fact a commonplace skill. I don't know of a composer of note (including contemporary) who can't do this -- even *I* can! It's something like learning to write English: when you were in 1st grade you probably had to say the words out loud as you wrote them. But with time and practice you didn't need to anymore --- you could hear the words in your head. Same with music.