[1] *She doesn't like [that I talked rudely to her].
[2] *She doesn't like [I talking rudely to her].
[3] She doesn't like [my talking rudely to her].
[4] She doesn't like [me talking rudely to her].
Preliminary point: the bracketed elements in [2]-[4] are not objects and the corresponding element in [1] is not a so-called 'noun clause'.
[1] is ungrammatical because the verb "like" does not license a content clause on its own. It requires "it", as in She doesn't like it that I talked rudely to her.
[2] is ungrammatical. Non-finite subordinate clauses take either genitive or accusative case subjects, but not nominative case.
In [3] and [4] the bracketed elements are subordinate clauses functioning as complement of "like". "My/me" is the subject and "talking rudely to her" is the predicate VP. The difference between the two is one of style. [3] has the more formal genitive case "my" as subject.