"work of art" is a common set phrase meaning an artistic creation such as a painting. "art of work" appears to mean work considered as art, but it is not a common phrase. Here it seems to be used for the sake of word play as much as anything else.
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Anonymousart of workIn expressions of the form "art of (something)", "art" is usually taken to be "skill" or "skillfulness". Thus, I take "art of work" as "craftsmanship", which is often called "technique" in the art world. Note that nevertheless "art of work" is not a standard term, and the expression has been invented for the sake of some clever word play