A common meaning of X with Y is being used to express "instrumentality". In this case, it's mostly equivalent to using the words "whenever X use(s)/used Y".
X is strict with punctuality = X is strict whenever X uses punctuality.
Punctuality is not something that is often "used" (like a tool or thing needed to complete an activity) so this would likely sound strange, but a possible valid meaning (and implication if you say this) would be if a manager or someone is known to "use" punctuality to find people to fire or get in trouble.
X about Y is more general than with and doesn't just mean "when X uses Y":
X is strict about punctuality = X is strict with anything X does that has to do with punctuality.
So, if you say something like "My manager is strict about punctuality" - you are saying that anything he/she does as a manager that also would deal with punctuality - he/she would be strict in whatever actions he/she would be doing. All the normal things a manager does in regard to punctuality - whatever they are, he/she is strict while doing those things.