"The exam is coming up! We need to start preparing."
I gave the same answer. I just pasted it in Google Translate and it came out exactly as, "The exam is coming soon." I reported it.
Sorry! That sounds unnatural. 要 is necessary to denote need, so it's not optional; without it, the latter half of the sentence reads as "we start to prepare." And if you put 開始 before 要, it sounds like you're saying "we start to need to prepare."