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

Answer: 
The sentence is correct. Here, "ill" is an *adverb* roughly meaning "badly", and it modifies "becomes".

See this [Merriam-Webster definition](https://www.merriam-webster.com/dictionary/ill):

> 
> **ill** adverb  
> 
> **worse; worst**  
> 
> **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*
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> 
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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"](https://www.merriam-webster.com/dictionary/becoming), meaning "suitable" or "fitting".

So your quoted sentence roughly means, "*Sarcasm like that **is a bad fit** for someone in Goldschmidt's position.*"
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*.