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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.

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 form of an imperative, without any indication of future tense, but it still refers to the future consequence of failing to do the first part. For example:

Be punctual or be eliminated.

Here are some very well-known examples:

Be there or be square. ["Square" here is old slang for "uncool", dull, undesirable, out of step with social life. Wiktionary.]

Sweat now or bleed later. [This means, if you don't work hard to build skills now, you'll suffer in some later situation where those skills will be needed, such as warfare.]