Is there a cardinal $\kappa \neq \emptyset$ and a connected poset $P$ of cardinality $\leq \kappa$ such that there is no surjective order-preserving map from $(\mathcal{P}(\kappa),\subseteq)$ onto $P$?
(We say that a poset is connected if it is connected as a directed graph.)
A trivial necessary condition for such a surjection to exist is that $P$ is bounded, and this is sufficient. For any set $X$, let $F(X)$ be the free bounded poset on $X$ (i.e., $F(X)=X\sqcup \{0,1\}$ with any two elements of $X$ incomparable). For any bounded poset $P$ with underlying set $X$, there is a canonical order-preserving surjection $F(X)\to P$, so it suffices to give an order-preserving surjection $\mathcal{P}(X)\to F(X)$. But this is easy; for instance, you can send any singleton to its unique element and any set with more than one element to $1$.
