The duality between filters and ideals can be seen as Zhen Lin describes, and thus answering your question negatively. A somewhat more positive answer (though only somewhat) is to notice that whatever category you concoct from filters, you can also concoct using ideals (in effect, using the duality between them). The resulting categories would then be isomorphic (not dual!).
Having said that, the answer to your question could turn out to be a positive one, I'm just not sure. The thing is that there are many ways to define the morphisms in a category whose objects are all pairs $(X,\mathcal F)$ where $X$ is a set and $\mathcal F$ is a filter on it. There seems to be quite a lot of flexibility on the choice of morphisms, so maybe one that suits your needs exists.
Just to clarify, there are two commonly considered categories of filters, described in Blass' article. These turn out to be very useful notions of categories (e.g., for one there is a natural notion of tensor product, the other is useful for constructive nonstandard analysis).