Is there a model theoretic realization of the concept of Arithmetical Hierachy?

There is one more well-known equivalence for $forall exists$ sentences.

Theorem (Chang-Los-Suszko). A theory $T$ is preserved under taking unions of increasing chains of structures if and only if $T$ is equivalent to a set of $forall exists$ sentences.

For a proof, see Keisler, "Fundamentals of model theory", Handbook of Mathematical Logic, p. 63.

I found a related paper, which is older and doesn't quite answer your question but may be of interest. R. C. Lyndon, "Properties preserved under algebraic constructions", Bull. Amer. Math. Soc. 65 n. 5 (1959), 287-299, Project Euclid

According to that paper, and MathSciNet, a general solution to your question should be contained in H. J. Keisler, "Theory of models with generalized atomic formulas", J. Symbolic Logic v. 25 (1960) 1-26,
MathSciNet, JStor