Where can we put primitive recursive functions in Chomsky hierarchy?
The Chomsky hierarchy concerns languages. Languages are total functions from $\Sigma^*$ to $\{0,1\}$, where $\Sigma$ is some non-empty finite set.

A language can be computed by a primitive recursive function if it can be computed by a total Turing machine which has a primitive recursive time upper bound. In particular, every language accepted by an LBA is primitive recursive, since we can simulate an LBA by a deterministic Turing machine running in double exponential time, and $2^{2^n}$ is primitive recursive.

In contrast, it is known that the Ackermann $A(n)$ function is decidable but not primitive recursive. Hence the language $\{ \langle n, A(n) \rangle : n \in \mathbb{N} \}$ is decidable but not primitive recursive.

Consequently, we find that primitive recursive languages sit between Type 0 and Type 1. In other words, every Type 1 language is primitive recursive, and every primitive recursive language is Type 0. Moreover, there are primitive recursive languages which are not Type 1 (for example, any EXPSPACE-complete language), and there are Type 0 languages which are not primitive recursive (for example, the graph of the Ackermann function).