I haven't done thorough research, but a brief search indicates the idiom "up for" something is from at least the 19th century. Meanwhile, the related idiom "down for" is much more recent, originating in 1935 jazz culture, but popularized in 1990s slang.
As with any idiom in any language, the older an expression, and the more widespread its use, the more it feels like a standard part of the language. "Up for something" feels standard, while "down for something" feels more like slang.
We can only speculate why some people started using a particular phrase that is the opposite of a current idiom -- for example, saying something is "bad" to mean it's "very good". Often this is because certain groups want to create a kind of "private dialect" hidden from authority figures. Young people want to hide their communication from their parents, criminals want to hide from police, oppressed minorities want to hide from their oppressors, and so on.
The negation is pretty straightforward:
I'm not up for that.
I'm not down for that.
"Ain't" is also used as a negation in various dialects:
I ain't up/down for that
but I would caution against using it without understanding the nuance. There is a fine line between imitation and mockery, and you might unwittingly offend someone.