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I'm hunting for the title of a short story where a senile old man appears in a capsule in a laboratory.

They eventually realize time is passing backwards for him. After years he helps them build the machine he's in, and he changes places with the him that has been aging normally in the timeline.

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Could this be Ian Watson's "The Very Slow Time Machine"?

Google Books offers this:

The Very Slow Time Machine arrives on earth in 1985. Its sole inhabitant is old and mad. Soon it becomes apparent that for him, time is going slowly backward. With every day, he is getting younger and saner. The world, and its whole concept of time, science and philosophy, must wait for him to speak. But while the world waits, it changes...

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