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I'm a native American English speaker, and I find it difficult to understand how "be punctual" can be made passive. Perhaps your book is only asking you to change the part after "or". You got that exactly right:

Be punctual or you will be eliminated by them.

By the way, what I am about to say might not be relevant to your assignment, but often, in a "do this or that will happen" sentence, we put the second part in the present tense, but it still refers to the future. For example:

Be punctual or be eliminated.