BUG: Index.insert raising when inserting None into new string dtype

doc/source/whatsnew/v2.1.2.rst

@@ -24,6 +24,7 @@ Bug fixes
 ~~~~~~~~~
 - Fixed bug in :meth:`DataFrame.resample` not respecting ``closed`` and ``label`` arguments for :class:`~pandas.tseries.offsets.BusinessDay` (:issue:`55282`)
 - Fixed bug in :meth:`DataFrame.resample` where bin edges were not correct for :class:`~pandas.tseries.offsets.BusinessDay` (:issue:`55281`)
+- Fixed bug in :meth:`Index.insert` raising when inserting ``None`` into :class:`Index` with ``dtype="string[pyarrow_numpy]"`` (:issue:`55365`)
 - Silence ``Period[B]`` warnings introduced by :issue:`53446` during normal plotting activity (:issue:`55138`)
 -
 

pandas/core/arrays/string_arrow.py

@@ -613,3 +613,8 @@ def _reduce(
             )
         else:
             return super()._reduce(name, skipna=skipna, keepdims=keepdims, **kwargs)
+
+    def insert(self, loc: int, item) -> ArrowStringArrayNumpySemantics:
+        if item is np.nan:
+            item = libmissing.NA
+        return super().insert(loc, item)  # type: ignore[return-value]

pandas/tests/indexes/base_class/test_reshape.py

@@ -54,6 +54,14 @@ def test_insert_datetime_into_object(self, loc, val):
         tm.assert_index_equal(result, expected)
         assert type(expected[2]) is type(val)
 
+    def test_insert_none_into_string_numpy(self):
+        # GH#55365
+        pytest.importorskip("pyarrow")
+        index = Index(["a", "b", "c"], dtype="string[pyarrow_numpy]")
+        result = index.insert(-1, None)
+        expected = Index(["a", "b", None, "c"], dtype="string[pyarrow_numpy]")
+        tm.assert_index_equal(result, expected)
+
     @pytest.mark.parametrize(
         "pos,expected",
         [