This appears to be a parody of the poem, "Against Idleness and Mischief", by Isaac Watts. See http://www.victorianweb.org/authors/rands/ajrbion1.html.
It was famously parodied by Lewis Carroll in Alice in Wonderland: http://www.online-literature.com/carroll/899/
In the original, the "busy little bee" "improve[s] each shining hour" by hard work. We generally talk about a "shining hour" meaning a time when something very useful or important was done. Like, "Publishing his first novel was Mr Jones shining hour." It's also possible that by "shining hour" Watts means hour of the daytime, i.e. when the sun is shining. Either interpretation makes sense in the poem.
So in the parody, the writer is trying to come up with something that would be the opposite of a "shining hour". Thus, a "shady hour". An hour when it is dark.