Question: > 
> Such **sarcasm ill becomes** anybody on the shaky ground that Goldschmidt here treads.
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Shouldn't *ill* be before *sarcasm*?

Answer: 
**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"](https://www.lexico.com/en/definition/become). 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*.