The inverse Ackermann function is primitive recursive.
One way to see this is to use the fact that a function $f$ is primitive recursive when and only when
- the graph of $f$ is primitive recursive, and
- $f$ is bounded above by some primitive recursive function.
The graph of the Ackermann function is primitive recursive, i.e. the characteristic function of the set $\lbrace \langle x, y, z \rangle : z = A(x,y)\rbrace$ is primitive recursive. This is because checking that $A(x,y) = z$ is easy once $x, y, z$ are given. One can always construct a table of all previous values of $A$ used to justify that $A(x,y) = z$. If $z$ is indeed the correct answer, then the code for this table is not much bigger than $\langle x, y, z\rangle$ (smaller than $17^{17^{x+y+z}}$, for example). So, given a proposed triple $\langle x, y, z \rangle$, we can search for the relevant table and determine whether or not $A(x,y) = z$ is true in a primitive recursive fashion. Of course, the Ackermann function is not bounded above by a primitive recursive function, but that is the only thing that goes wrong.
Since the graph of the Ackermann function is primitive recursive, then so is the graph of the inverse Ackermann function $Ack^{-1}(z) = \max\lbrace x : A(x,x) \leq z\rbrace$. Moreover, the growth rate of $Ack^{-1}$ is bounded by some primitive recursive function (e.g. the identity function). It follows that $Ack^{-1}$ is indeed primitive recursive.
(This is more of a comment than an answer, but it's a bit too long to be split into comments so I'll post it as an answer.)
I don't know about functions defined on an arbitrary admissible set, but at least for admissible levels of the constructible hierarchy, you might be interested in what are called "$(\infty,0)$-recursive functions" in chapter VIII ("Recursion on Ordinals") of Peter G. Hinman's book Recursion-Theoretic Hierarchies (1978, available here), and also on this related question I asked a while ago while trying to make sense (without much success) of the various definitions. Hinman writes (op.cit., p.378) that:
The $(\infty,0)$-recursive functions will play somewhat the role here of the primitive recursive functions of ordinary recursion theory.
Perhaps even more relevant to your question would be the $(\infty,\lambda)$-recursive functions in Hinman's terminology, or even more the primitive $(\infty,\lambda)$-recursive functions in the terminology of the question I linked to, where $\lambda$ is the height of the admissible set considered (at least for a $L_\lambda$). But as I noted, the precise relation between these concepts escapes me.
Also somewhat relevant to your question might be Stephen G. Simpson's paper titled "Short Course on Admissible Recursion Theory" on p. 355–390 of Fenstad, Gandy & Sacks (eds.), Generalized Recursion Theory II (1978), proceedings of a symposium held in Oslo in 1977. It contains the clearest (if terse) explanation I found so far of how primitive recursive ordinal functions are defined and how they relate to more general recursion on ordinals.
Best Answer
You can express the totality of any computable function in PRA, using Kleene's T predicate, which is primitive recursive. So if you pick any index $e$ for the Ackermann function, the formula $(\forall n)(\exists t) T(\underline{e}, n, t)$ is already in the language of PRA.
However, you cannot prove the totality of the Ackermann function in PRA. One way to see this is to note that PRA is a subtheory of $\text{I-}\Sigma^0_1$, modulo an interpretation of the language of PRA into $\text{I-}\Sigma^0_1$. The provably total functions of $\text{I-}\Sigma^0_1$ are well-known to be exactly the primitive recursive functions.
There is a lot of proof theory literature on provably total functions, which are also called provably recursive functions. But I don't know how much of it focuses specifically on primitive recursive arithmetic. One place to look might be Hájek and Pudlák, Metamthematics of First-Order Arithmetic.