What do you call a person that criticizes your ignorance, but does nothing to help alleviate it?

You can say Bob is or behaves like a know-it-all.

know-it-all

noun

a person who behaves as if they know everything.

His debate style could be described as beating around the bush.

beat around the bush

to speak evasively or misleadingly, or to stall or waste time

which describes Bob's demeanour, criticizing Alice for not knowing the purpose of the umbrella, yet evading the explanation of it to her.

criticaster

http://dictionary.reference.com/browse/criticaster?s=t

The dictionary merely states that a criticaster is "an incompetent critic," but there's a lot more to it. It's a pretty intriguing word. It contains the right kind of derisive note, sufficiently high-pitched to be distinct in any circumstances, yet not harsh enough truly to injure.