$\newcommand{\cat}{\mathrm} \newcommand{\St}{\cat{Stone}^\cat{fr}} \newcommand{\Cat}{\cat{Cat}} \newcommand{\Cart}{\cat{Cart}} \newcommand{\Fun}{\cat{Fun}} \newcommand{\Mon}{\cat{Mon}} \newcommand{\Set}{\cat{Set}} \newcommand{\Po}{\cat{Poset}}$ EDIT : I misread the definition of ultracategory fibration, apparently only certain locally cartesian edges are closed under composition. In particular ultracategory fibrations are not cartesian fibrations, and so most of what I said is incorrect - at least the.proofs I'm unsure about the results
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)