In linguistics, a case is how a noun declines with respect to its grammatical function within a given phrase, clause, or sentence. Is there a linguistics term to refer to the “state” of a noun within a given sentence? (For example, it can be the subject, or the object, etc.) Or is it simply referred as the state of a noun?
You're perhaps looking for the term
Part of Speech. It's a rather vague term, and in syntax theory you usually wanna specify the noun by stating its
thematic role (agent, patient, experiencer, theme, etc.) and its
syntactic function (subject, object, etc.). Theories like LFG (Bresnan, doi.org/10.1002/9781119105664) have "subject" as an atomic concept (a "function"), but Bare Phrase Structure (aka Minimalism, Chomsky http://dx.doi.org/10.7551/mitpress/9780262527347.001.0001) does not.