Such sarcasm ill becomes anybody on the shaky ground that Goldschmidt here treads.
Shouldn't ill be before sarcasm?
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The sentence is correct. Here, "ill" is an adverb roughly meaning "badly", and it modifies "becomes".
See this Merriam-Webster definition:
1 a : in a faulty, inefficient, insufficient, or unpleasant manner —often used in combination
// the methods used may be ill-adapted to the aims in view
It's also worth noting that the word "become" has a far less common meaning as well, which is in the sense of the adjective "becoming", meaning "suitable" or "fitting".
So your quoted sentence roughly means, "Sarcasm like that is a bad fit for someone in Goldschmidt's position."
ill becomes is an idiomatic phrase that means "is not suitable for", or "is not appropriate for"; an old-fashioned meaning for become is "to be appropriate or suitable; to look good on". So the sentence means something like "Sarcasm like this is not appropriate for anybody..."
Changing it to "Such ill sarcasm becomes anybody..." doesn't make sense. It makes ill into a modifier for sarcasm, which isn't what's intended. Also, "Such ill sarcasm becomes ____" either means the sarcasm is turning into something else (which completely changes the meaning) or the sarcasm is appropriate, which reverses the meaning of the sentence!
To build on what @stangdon has said:
If you were to place ill before sarcasm then you would using ill as an adjective and applying it to the word sarcasm. Essentially you would be writing Ill-Sarcasm: sarcasm used in an inappropriate way (which is understandable to a native speaker, but is not grammatically correct).
This would completely change the meaning of the sentence.
"Such sarcasm ill becomes anybody on the shaky ground that Goldschmidt here treads." means: sarcasm doesn't suit somebody on shaky ground
"Such ill-sarcasm becomes anybody on the shaky ground that Goldschmidt here treads." means: ill-sarcasm does suit somebody on shaky ground.