diff --git a/doc/source/whatsnew/v2.2.0.rst b/doc/source/whatsnew/v2.2.0.rst
index 70039cc697b8a..cda4da9d76c42 100644
--- a/doc/source/whatsnew/v2.2.0.rst
+++ b/doc/source/whatsnew/v2.2.0.rst
@@ -604,6 +604,7 @@ Strings
 - Bug in :meth:`Series.str.find` when ``start < 0`` for :class:`ArrowDtype` with ``pyarrow.string`` (:issue:`56411`)
 - Bug in :meth:`Series.str.replace` when ``n < 0`` for :class:`ArrowDtype` with ``pyarrow.string`` (:issue:`56404`)
 - Bug in :meth:`Series.str.startswith` and :meth:`Series.str.endswith` with arguments of type ``tuple[str, ...]`` for ``string[pyarrow]`` (:issue:`54942`)
+- Bug in comparison operations for ``dtype="string[pyarrow_numpy]"`` raising if dtypes can't be compared (:issue:`56008`)
 
 Interval
 ^^^^^^^^
diff --git a/pandas/core/arrays/string_arrow.py b/pandas/core/arrays/string_arrow.py
index 32ab3054c0f51..50cd052f80abd 100644
--- a/pandas/core/arrays/string_arrow.py
+++ b/pandas/core/arrays/string_arrow.py
@@ -41,6 +41,7 @@
     BaseStringArray,
     StringDtype,
 )
+from pandas.core.ops import invalid_comparison
 from pandas.core.strings.object_array import ObjectStringArrayMixin
 
 if not pa_version_under10p1:
@@ -662,7 +663,10 @@ def _convert_int_dtype(self, result):
         return result
 
     def _cmp_method(self, other, op):
-        result = super()._cmp_method(other, op)
+        try:
+            result = super()._cmp_method(other, op)
+        except pa.ArrowNotImplementedError:
+            return invalid_comparison(self, other, op)
         if op == operator.ne:
             return result.to_numpy(np.bool_, na_value=True)
         else:
diff --git a/pandas/tests/series/test_logical_ops.py b/pandas/tests/series/test_logical_ops.py
index 153b4bfaaf444..d9c94e871bd4b 100644
--- a/pandas/tests/series/test_logical_ops.py
+++ b/pandas/tests/series/test_logical_ops.py
@@ -530,3 +530,19 @@ def test_int_dtype_different_index_not_bool(self):
 
         result = ser1 ^ ser2
         tm.assert_series_equal(result, expected)
+
+    def test_pyarrow_numpy_string_invalid(self):
+        # GH#56008
+        pytest.importorskip("pyarrow")
+        ser = Series([False, True])
+        ser2 = Series(["a", "b"], dtype="string[pyarrow_numpy]")
+        result = ser == ser2
+        expected = Series(False, index=ser.index)
+        tm.assert_series_equal(result, expected)
+
+        result = ser != ser2
+        expected = Series(True, index=ser.index)
+        tm.assert_series_equal(result, expected)
+
+        with pytest.raises(TypeError, match="Invalid comparison"):
+            ser > ser2