In 1896, the Oxford English Dictionary described this word, when its entry for it was first published, as “humorously pedantic”. The entry defined it as a verb meaning to cudgel or beat.
These days fustigate is mainly fodder for lists of difficult or rare words. But then the word hasn’t had a particularly extensive or distinguished history — it only came into the language around 1650 and even in its prime it was always rather an uncommon or literary word. Its creators took it from the Latin verb fustigare, to cudgel to death (from fustis, a staff or club).
Sir Richard Burton employed it in the 1880s in his translation of The Book of the Thousand Nights and a Night (better known as the Arabian Nights Entertainment): “And she bade them bash me; so they beat me on my ribs and the marks ye saw are the scars of that fustigation.”
A writer in the Living Age in 1896 used it figuratively for severely criticising a person or a thing. He wrote of Matthew Arnold’s “fustigation of dummy opponents” as part of his style. This figurative sense survives to some extent:
Actually, most of today’s complaints seem weak and whiny, almost apologetic. They lack the scorn and vitriol the writers evidently feel in their hearts. So — just this once, at the dawning of a new year — let me pass along a sample of real fustigation from real experts, a no-nonsense, in-your-face style for local critics to aim for.
Rocky Mountain News, 17 Dec. 2004.