Your superscripts are all off by one with respect to the usual notation. That is, $\Sigma^0_n$ in the language of set theory means we have first-order quantifiers only, that is, over sets, whereas $\Sigma^1_n$ means we have $n$ alternations of second-order quantifiers (over classes).
If one is speaking of first-order formulas and set objects only, then the property in a model of set theory that every definable class $\{x\mid\varphi(x)\}$ has a definable element is equivalent to $V=\text{HOD}$, because the class of non-OD sets is certainly definable, but can have no definable element, and so it must be empty if the property is to hold; conversely, if $V=\text{HOD}$ holds, then there is a definable well-ordering of the universe, and so the definable class has a least member with respect to that well-order, and this object will be definable.
So the models of set theory in which every definable class has a definable element are exactly the models of V=HOD.
But you seem to be interested not in defining classes of sets, but rather in properties of classes $C$. That is, if $\varphi$ is a property of a class $C$ and there is a class $C$ such that $\varphi(C)$, then is there a definable class $C$ for which $\varphi(C)$?
Here, the answer is: not necessarily. For example, consider a model of NGBC that has a truth-predicate $T$ for first-order truth. For example, any model of Kelley-Morse set theory is like that, but considerably less than KM suffices. Now, the assertion that "$T$ is a truth predicate" is a first-order expressible property of $T$, since one need only say that $T$ obeys the Tarskian recursion. But there can be no definable class $T$ that is a truth predicate, because of Tarski's theorem on the non-definability of truth.
Meanwhile, there are models of NGB set theory that are pointwise definable, meaning that every set and class is definable without parameters. For example, see my paper:
- Hamkins, Joel David; Linetsky, David; Reitz, Jonas, Pointwise definable models of set theory, J. Symb. Log. 78, No. 1, 139-156 (2013), http://dx.doi.org/10.2178/jsl.7801090. (ZBL1270.03101).
In such a model, if there is a class $C$ with property $\varphi(C)$, then $C$ is already definable, because all classes in such a model are definable.
So the answer is that it is sometimes true, and sometimes not true, depending on your model of NGB.