$\newcommand{\cat}{\mathrm}
\newcommand{\St}{\cat{Stone}^\cat{fr}}
\newcommand{\Cat}{\cat{Cat}}
\newcommand{\Cart}{\cat{Cart}}
\newcommand{\Fun}{\cat{Fun}}
\newcommand{\Mon}{\cat{Mon}}
\newcommand{\Set}{\cat{Set}}
\newcommand{\Po}{\cat{Poset}}$
EDIT : I misread the definition of ultracategory fibration, apparently only certain locally cartesian edges are closed under composition. After a thought about it, what I said for monoids remains true, but not for posets. I just need to modify the arguments.
I would argue that a monoid is not the same as a category with 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.
The first question is : What is a pointed ultracategory ?
In my original post, I made a mistake because I had misread the definition of ultrcategory. In particular, a morphism $\St\to \mathcal M$ is more than just a point in $\mathcal M_*$ : it's a point with a certain property. Namely, (see definition 1 in Lurie's notes) we ask that the comparison morphisms for base-change functors associated to all compositions $\beta I\to \beta J\to \beta K$ be isomorphisms when applied to this point (while in the definition of an ultracategory, one only requires this if $\beta I \to \beta J$ comes from a map $I\to J$).
But, in particular, if this point is the only point of $\mathcal M_*$ (and thus of $\mathcal M_{\beta I}$ for all $I$), this amounts to asking that cartesian morphisms actually be closed under composition ! In other words,
An ultramonoid is an ultracategory which is a cartesian fibration $\mathcal M\to \St$ with a section $\St\to \mathcal M$ sending cartesian edges to cartesian edges, such that on fibers over $*$, the map $*\to \mathcal M_*$ is essentially surjective.
Now that this is clarified, the rest of my answer essentially goes through without changes. But note that there was initially a mistake, and also note that this will not work for posets !
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.
By what was clarified above, ultramonoids can therefore be viewed as a full subcategory of $\Fun((\St)^\cat{op},\Cat_*)$, via the Grothendieck construction (while this is not the case for general ultracategories ! for them it would be certain pseudo-functors). 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.
(note that for ultrasets the subtlety disappears: the fibrations must be cartesian, and not only locally cartesian)
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.
Now, if $\mathcal M$ is an ultracategory with a point $x : *\to \mathcal M_*$, there is, in general, no way to make an ultramonoid out of it because of this cartesian vs locally cartesian business. However, if it is a "nice" point, i.e. if it is a point for which the comparison morphisms for general compositions $\beta I\to \beta J\to \beta K$ are isomorphisms, then we can view it as a morphism $\St\to \mathcal M$ of ultracategory fibrations. In particular we can take the full subcategory $\mathcal M_x$ of $\mathcal M$ spanned by the image of $\St$, and this should still be an ultracategory fibration. But now this is a pointed ultracategory fibration with an essentially surjective point, so it is an ultramonoid, as explained above.
We therefore find tha a sufficiently nice object $x$ (in more details: a point $\St\to \mathcal M$) in an ultracategory $\mathcal M$ has an ultramonoid of endomorphisms, $\cat{End}_\mathcal M(x)$, i.e. a compact Hausdorff topological space of endomorphisms.
(For posets, I don't know a satisfying answer. I'm not sure you will get a more satisfying answer than to the question "what is an ultracategory ?", but I would be interested in seeing one)
EDIT to answer the questions in a comment below (rather than as a long sequence of further comments):
1- I used "pointed category" to mean something more naive than in the nLab, namely a category $C$ with a point $*\to C$. Morphisms of such are functors $f : C\to D$ that preserve the point, and $2$-morphisms are natural transformations that are the identity on the point (this is an unfortunate clash of terminology: one can define a "pointed object" in an arbitrary category, and "pointed objects of $Cat$" do not coincide with the nLab's pointed categories). With this definition, the (usual) $1$-category of monoids is equivalent to the $(2,1)$-category of pointed categories for which the point $*\to C$ is essentially surjective.
2- "which question": I was adressing the question about ultramonoids (at first I thought also posets, but in the end, no). I don't know if I'm using the same definition of ultramonoid as you are, because it's unclear to me what "an ultracategory with one object" means in the same way that it's unclear to me what "a category with one object" means.
If you mean in the strictest sense and using Definition 7, then no I am not using the same definition (a choice which my first paragraph tries to explain). There are 2 reasons I did not use the same definition: a- what I explained in the first paragraph of this (partial) answer and b- the strict definition is not invariant under the several presentations of "ultracategories" and not invariant under equivalence of ultracategories. I know that this means I'm not technically answering your question (in any case, I wasn't answering all of it, this was just a contribution ! I'm sorry if I made it sound like it was supposed to be everythin). But maybe in the situation you're interested in, the definition I used might be more relevant ?
3- The notation $*_{\beta I}\to \mathcal M_{\beta I}$ refers to $*^I\to \mathcal M^I$ in the language of definition 7, specifically the map that picks out this one object. Sorry again, Definition 1 was clearer to me, which is why I phrased my answer in this language (note that the two definitions are equivalent in a suitable sense).
4- I define the $(2,1)$-category of ultramonoids as a full subcategory of the category of pointed ultracategories (in the sense I described earlier). In particular, morphism categories (rather than morphism sets) are categories of ultrafunctors and, I guess but am not sure about the terminology, ultra natural transformations.
5- A "point" of a category is an object therein, equivalently a functor $*\to C$. I used this latter perspective (which may seem unnecessarily pedantic) to define a point of an ultracategory, as an ultrafunctor from the trivial ultracategory $*$ to $\mathcal M$. That's the content of the paragraph following "The first question is..." - I was trying to identify those more concretely.
6- I'm not sure what the nice objects are in the case of "models of a first order theory". Essentially these are the models $M$ such that for any two sets $J,K$, function $f: J\to \beta K$ and ultrafilter $\mathcal U$ on $J$, the canonical morphism $(\prod_K M)/\mathcal V \to \prod_J ((\prod_K M)/\mathcal f(j))/\mathcal U$ is an isomorphism, where $\mathcal V = \lim_\mathcal U f(j)$ in $\beta K$, equivalently $\mathcal U\wr \mathcal V_\bullet$ where $\mathcal V_\bullet = f$.
It looks like this map is always injective by definition of $\mathcal U\wr \mathcal V_\bullet$, but most likely almost never surjective. For instance take $K =*$, then this is $\mathcal M\to (\prod_J \mathcal M)/\mathcal U$, which is almost never surjective.
7- I'm sorry, I misunderstood that these were your main questions. So, for the question of which structures, this amounts to answering 6-. The answer seems to be "almost never" - my guess would be "if and only if $M=*$". It makes sense to a certain extent : what would be the compact Hausdorff topology on the set of elementary embeddings/morphisms ?
For the second question however, the way everything is set up makes it very easy to answer : a good point in $\mathcal M$ is an ultrafunctor $*\to \mathcal M$. This is clearly stable under composition of ultrafunctors : if $F:\mathcal{M\to N}$ is an ultrafunctor, so is $*\to \mathcal M\to\mathcal N$, and in particular, $F$ preserves "nice" objects. A corollary of how everything is set up is also (for free) that you get a morphism of ultramonoids (in the sense I described, so a morphism of compact Hausdorff topological monoids) $End(M)\to End(F(M))$.
8- By strongly connected I meant "only one isomorphism class". I'm not sure there is a standard terminology and what it is, if there is.
