I was thinking about this question and I have my own thoughts on this, please do let me know if my reasoning is correct.
The question asks if I can pick an $X$ such that this condition is fulfilled for all $Y$.
If the function $f$ is to be surjective, it must mean that every element must be an output of $f$. So if $X$ was countably infinite, this would not be possible, since I can pick $Y$ an uncountably infinite set. So $X$ would have to be uncountably infinite set, but then this set $X$ would not work if $Y$ was the null set, because a function only exists from $f:X \rightarrow \emptyset $ only if $X$ is the null set.
So this statement is false. Is my reasoning correct?