Okay, let me try to summarize.

The fact that a finite join semilattice with a minimum is a lattice is Proposition 3.3.1 of Stanley, "Enumerative Combinatorics," Volume 1.

Now suppose you have a finite lattice $L$ (corresponding to your $\mathcal{O}(P)$, but the fact that it is distributive is irrelevant), and another poset $K$ for which you have an order-preserving, join-preserving, surjective map $h\colon L \to K$. This does imply that $K$ is a join semilattice (for $a,b \in K$, we define the join $a\vee b$ to be $h(a' \vee b')$ where $a', b' \in L$ are pre-images under $h$ of $a,b$). Furthermore, the image of the minimum of $L$ under $h$ must be the minimum of $K$. And clearly $K$ is finite. So indeed, we can conclude that $K$ is a lattice.

Okay, let me try to summarize.

The fact that a finite join semilattice with a minimum is a lattice is Proposition 3.3.1 of Stanley, "Enumerative Combinatorics," Volume 1.

Now suppose you have a finite lattice $L$ (corresponding to your $\mathcal{O}(P)$, but the fact that it is distributive is irrelevant), and another poset $K$ for which you have an order-preserving, join-preserving, surjective map $h\colon L \to K$. This does imply that $K$ is a join semilattice (for $a,b \in K$, we define the join $a\vee b$ to be $h(a' \vee b')$ where $a', b' \in L$ are pre-images under $h$ of $a,b$). Furthermore, the image of the minimum of $L$ under $h$ must be the minimum of $K$. And clearly $K$ is finite. So indeed, we can conclude that $K$ is a lattice.

EDIT: In response to the further question about whether $h$ must be meet-preserving, very simple examples show this is not the case. For instance, take $L$ to be the rank $2$ Boolean lattice of subsets of $\{1,2\}$, and $K$ to be the two element lattice $\hat{0} < \hat{1}$, with $h(\{1\})=h(\{2\})=h(\{1,2\})=\hat{1}$ and $h(\varnothing)=\hat{0}$. This $h$ is order-preserving, join-preserving, and surjective, but not meet-preserving.