"The exam is coming up! We need to start preparing."
I don't think it does! Pretty sure the word order is supposed to go subject-adverb-verb-object. The only way this sentence makes any sense to me is if what we think is the verb is actually a description of when it will happen. Stuff about time can go at the very start or after the subject.
I think that in 快要考试了, the term 考试 is being used as a verb, which is why it can be used in the 快要...了 structure.
There seems to be some abiguity about 考试. The dictionary nciku.com only classifies 考试 as a verb (which it defines as "sit an exam"). On the other hand, Google translate only classifies 考试 as a noun (meaning "examination," "exam," or "test.") I was taught that 考试 is a verb+object construction (like 跑步, 跳舞, 睡觉, etc).