No, that's not the way to view it. In order to view a monoid as a category, you have a single object $\mathsf{Andreas}$, and each element of the monoid is one morphism $\mathsf{Andreas}\to\mathsf{Andreas}$ in the category. The monoid operation is the composition in the category.
So for the integers, you don't have a morphism "add 1", but a morphism that is simply called $1$. And composition in the category works such that $1$ composed with $1$ is the morphism called $2$.
This is an example of a category where the morphisms are not functions.
In algebra, a monoid is defined as a pair $(M, *)$ where $M$ is a set and $*$ is a binary operation on $M$ satisfying associativity and having an identity element.
A category with a single object, on the other hand, is given by a triple $(O, M, *)$ where $O$ is a set with a single element (the class of objects), $M$ is a set (the set of morphisms from the single object to itself is the only hom-set) and $*$ is a binary operation on $M$ satisfying associativity and having an identity element (the composition of morphisms turns out to be a binary operation because there is only one hom-set).
Notice that in the definition of a category with a single object there is no relationship between $O$ and $(M, *)$, because the properties that need to hold involve only $(M, *)$. Since those properties are exactly the same as those in the definition of monoid, we can say that a category with a single object consists of a monoid together with a set with a single element.
Therefore, if you want to get a monoid from a category with a single element, you just have to discard the useless set with a single element. Conversely, if you want to get a category with a single element from a monoid, you just have to come up with a set with a single element. Since that element won't play any role in the category, one usually puts a placeholder like $@$ to mean that any object will do.
Two categories with a single element are isomorphic (not just equivalent) whenever the corresponding monoids are isomorphic. Indeed, if $f\colon M \to M'$ is a monoid isomorphism, then the functor $F \colon \mathcal C \to \mathcal C'$ defined by $F(@)=@'$ and $F(m) = f(m)$ for any $m \in M$ (that is $m \colon @ \to @$ as a morphism) is clearly an isomorphism of categories, its inverse $F^{-1} \colon \mathcal C' \to \mathcal C$ being given by $F^{-1}(@') = @$ and $F^{-1}(m')=f^{-1}(m')$ for any $m' \in M'$ (i.e., $m' \colon @' \to @'$).
Now, if you still think that it makes more sense to define a monoid as a category with a single object, and that in order for a monoid defined in this way to be interesting its object should be some algebraic object as well, then you could consider the monoid of endomorphisms of a given algebraic object.
For example, if $(R, +, *)$ is a ring, then you could consider the full subcategory of the category of rings having $(R, +, *)$ as its only object. In this way there is indeed a relationship between the object and the morphisms, since the morphisms are precisely the ring homomorphism from $(R, +, *)$ to itself.
In fact, this construction can be applied to any category, not just a category of algebraic objects (e.g., you could also consider the full subcategory of the category of topological spaces having a given topological space as its only object, and so on).
Best Answer
It doesn't make a difference what the object is, so I doubt anybody has ever specified what it is. This is the same kind of thing as when you consider a "one-point space" $\{*\}$ in topology.
You said that a morphism is determined by its domain and codomain. (Or source and target object, so to speak.) That's not true, even in the category of sets. There's usually more than one mapping from a set $A$ to a set $B$.
For the last thing, what you wrote is correct. Of course $1$ and $2$ are not true "mappings," but they're what stands in for mappings in this category. If $1$ and $2$ are thought of as analogous to mappings, then the operation $+$ should be thought of as analogous to $\circ$. So $1 \circ 2 = 3$, etc.
By the way, this category has a concrete realization as the category with a single object $\mathbf{Z}$ whose morphisms are all automorphisms of $\mathbf{Z}$ as an ordered set.