You could use either 'like' or 'unlike' in that sentence and probably be understood, but the punctuation and/or phrasing could be better. The way it's written it's unfortunately ambiguous, but a native speaker would likely understand that the cat didn't pee, whereas most people would have.
When speaking, the distinction would be made by a pause.
With a pause, indicated in writing with a comma:
He didn't even pee himself, unlike a lot of people do the first time they jump.
I would probably leave out the 'do' in this version. I think the reason for this is that while 'like' can be used as an adverb, 'unlike' generally isn't. 'Do' is acting as a pro-verb for 'pee himself' (which would be 'pee themselves' since the subject is 'people'). So, substituting for 'do', we have:
He didn't even pee himself, unlike a lot of people pee themselves the first time they jump.
Which sounds pretty awkward, as 'unlike' - an adjective - is comparing two actions, peeing oneself and not peeing oneself. Removing 'do', we have:
He didn't even pee himself, unlike a lot of people the first time they jump.
In which case 'unlike' - an adjective - is now comparing a cat that didn't pee with people who did, which are both nouns.
Without a pause:
He didn't even pee himself like a lot of people do the first time they jump.
Here, the do is optional, as 'like' is comfortable being either an adverb or an adjective. For completeness:
He didn't even pee himself like a lot of people the first time they jump.
My Preference
For what it's worth, I prefer the version with 'unlike', a pause, and no 'do'. I think this does the best job of clearly communicating the contrast.
He didn't even pee himself, unlike a lot of people the first time they jump.