The statements, along with the literary device used in each, are below:
“The Army Alpha is cleverly designed"
-- This is an example of understatement. By describing the Army Alpha as cleverly designed, the speaker is downplaying the scope of the test. We are told the told the test will show the Army everything about you. Therefore, it is an understatement to say it is "cleverly designed."
“I guess that the letter was the last straw”
-- This is an example of an idiom. An idiom is a figure of speech whose meaning seems unrelated to the words used. The phrase "the last straw" has nothing to do with straws: it is a figure of speech used to explain when someone has run out of patience, etc. Therefore this is an example of an idiom.
“I took the opportunity to vomit out a Niagara of vitriol”
-- This is an example of hyperbole. Hyperbole is extreme exaggeration. Since it is impossible to vomit a literal waterfall of anything, this statement is hyperbole.
“Army Intelligence called me up in about a week, wanting to chat”
-- This is an example of irony. Army Intelligence is not calling for a chat. The speaker is being sarcastic, or ironic. Army Intelligence is going to interrogate the speaker in the presence of his lawyer. A "chat" this is not.