$\newcommand{\cat}{\mathrm}
\newcommand{\St}{\cat{Stone}^\cat{fr}}
\newcommand{\Cat}{\cat{Cat}}
\newcommand{\Cart}{\cat{Cart}}
\newcommand{\Fun}{\cat{Fun}}
\newcommand{\Mon}{\cat{Mon}}
\newcommand{\Set}{\cat{Set}}
\newcommand{\Po}{\cat{Poset}}$

I would argue that a monoid is not the same as a category woth one object, namely a monoid is the same as a *pointed* category with one object. This is not really relevant if you look at the category of categories as a $1$-category (because then $\hom_\Cat(BM, BM')$ is indeed isomorphic to $\hom_\Mon(M,M')$), but it does if you more naturally view it as a $(2,1)$-category ($\Fun(BM,BM')$ has nontrivial morphisms !)

In particular, I'll interpret your question as : 

> What is an ultracategory $\mathcal M$ with a morphism from the terminal ultracategory "$*$" such that for all $I$, $*_{\beta I}\to \mathcal M_{\beta I}$ is essentially surjective ?

Of course, because $*_{\beta I}\simeq \prod_I*$ in a way compatible with $\mathcal M_{\beta I}\simeq \prod_I\mathcal M_*$, this is equivalent to $*\to \mathcal M_*$ being essentially surjective. Also, because $(\St)^\cat{op}$ has an initial object (the terminal object of $\St$, namely $*$), a pointed ultracategory is just an ultracategory $\mathcal M$ with a point in $\mathcal M_*$.

The second thing to observe is that there is a functor $\Mon\to\Cat_*$ which implements the equivalence of $(2,1)$-categories I've mentioned above. 

This functor is fully faithful (in fact it has a right adjoint - given by $(C,x)\mapsto \cat{End}_C(x)$, this will be relevant later - and the unit map $M\to \cat{End}_{BM}(\bullet)$ is an isomorphism), and therefore the functor $\Fun((\St)^\cat{op}, \Mon)\to \Fun((\St)^\cat{op},\Cat)$ is also fully faithful. 

Note that ultracategories,  viewed as ultracategory fibrations (Definition 1. in the linked notes), can be seen as a full subcategory of $\Fun((\St)^\cat{op},\Cat)$, via the Grothendieck construction. Precisely, it is the full subcategory spanned by functors $(\St)^\cat{op}\to\Cat$ that send the diagrams $(\{i\}\to \beta I)_{i\in I}$ to product diagrams. Piecing things together, using the fact that $\Mon\to\Cat_*$ and $\Cat_*\to\Cat$ both reflect and preserve products, we find : 

> The category of ultramonoids is equivalent to the full subcategory of $\Fun((\St)^\cat{op},\Mon)$ spanned by those functors that send the diagrams $(\{i\}\to \beta I)_{i\in I}$ to product diagrams.

But we can be more concrete. Indeed, $\Fun((\St)^\cat{op},\Mon)\simeq \Mon(\Fun((\St)^\cat{op},\Set))$, and $\Mon \to \Set$ preserves and reflects products, so that : 

> The category of ultramonoids is equivalent to the category of monoids in the category of ultrasets. 

Ultrasets are compact Hausdorff topological spaces, so finally: 

**Corollary** : The category of ultramonoids is equivalent to the category of monoids in compact Hausdorff topological spaces. 

Note that if $\mathcal M$ is an ultracategory with a point $x : *\to \mathcal M_*$ we can deduce from it a pointed ultracategory $*\to \mathcal M$ as explained above, and therefore by applying the right adjoint to $\Mon\to\Cat_*$, $\cat{End} : \Cat_*\to \Mon$, we find that indeed, an object $x$ in an ultracategory $\mathcal M$ has an ultramonoid of endomorphisms, $\cat{End}_\mathcal M(x)$, i.e. a compact Hausdorff topological space of endomorphisms. 

Finally, let us move to posets - I will provide less details, the ideas are mainly the same. Now, the category $\Po$ of posets (I mean partially ordered sets here - if you mean preordered, then you have to add $2$-morphisms) is a full sub-$(2,1)$-category of $\Cat$, and so exactly in the same way as above, we can view ultraposets as certain functors $(\St)^\cat{op}\to \Po$. The condition "for all $x,y$, $|\hom_C(x,y)|\leq 1$ is also closed under products. 

Note that a poset can be described as a set $P$ with a monomorphism $R\to P\times P$ satisfying certain conditions that can be described in terms of limits. Specifically, we have (reflexivity) $\Delta_P : P \to P\times P$ factors through $R$, (transitivity) $R\times_P R\to P\times P$ factors through $R$, (antisymmetry) the factorization $P\to R\times_{P\times P}R^\cat{op}$ is an isomorphism.

(note that "factors through" is a limit condition for monomorphism : given a monomorphism $A\to B$ and an arrow $X\to B$, "$X\to B$ factors through $A$" can be expressed as $X\times_B A\to X$ is an isomorphism - indeed, it is a monomorphism, thus an isomorphism if and only if it admits a section, if and only if there is a factorization)

In particular, just as before, we can describe ultraposets as the same limit description but in ultrasets, i.e. in compact Hausdorff topological spaces. 

**Corollary** : Ultraposets are posets in compact Hausdorff topological spaces. 

(where a poset in a category $C$ with finite limits is an object $P$ with a monomorphism $R\to P\times P$ satisfying the axioms outlined above)