Bjorn's answer is not quite correct because the quantifier for $C$ is $\exists$. But it can be fixed. Take a partition of $\mathbb N$ into 3-element subsets $\{3n+1,3n+2, 3n+3\}$, $n=0,1,...$. Then let $C$ be the set of numbers not divisible by 3. Take a non-principle ultrafilter containing $C$ (it exists by the Zorn lemma). It is a counterexample: none of the partition classes is in the ultrafilter, because it is non-principle, and $C$ intersects each partition class by 2 elements.
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