I'm going to pass the exam soon. You need the definite article. You would only use "I'm going to pass exam soon" if you were texting, where you want to be as short as possible. Otherwise, you should take the trouble to use articles such as 'the','a' and 'an'.

Is this sentence correct? I'm going to pass the exam soon

I think you must say :I am going to pass an exam soon

I think it is not correct. You can say : I am going to take an exam.OR: I am going to have an exam. To pass an exam means to succeed .