diff --git a/doc/source/whatsnew/v1.0.4.rst b/doc/source/whatsnew/v1.0.4.rst
index 95007f4dd1caf..4020dfc50150a 100644
--- a/doc/source/whatsnew/v1.0.4.rst
+++ b/doc/source/whatsnew/v1.0.4.rst
@@ -26,6 +26,7 @@ Fixed regressions
 - Fix regression in :meth:`DataFrame.describe` raising ``TypeError: unhashable type: 'dict'`` (:issue:`32409`)
 - Bug in :meth:`DataFrame.replace` casts columns to ``object`` dtype if items in ``to_replace`` not in values (:issue:`32988`)
 - Bug in :meth:`GroupBy.rolling.apply` ignores args and kwargs parameters (:issue:`33433`)
+- More informative error message with ``np.min`` or ``np.max`` on unordered :class:`Categorical` (:issue:`33115`)
 - 
 
 .. _whatsnew_104.bug_fixes:
diff --git a/pandas/compat/numpy/function.py b/pandas/compat/numpy/function.py
index 7158f251ad805..9cae127a0a549 100644
--- a/pandas/compat/numpy/function.py
+++ b/pandas/compat/numpy/function.py
@@ -209,7 +209,7 @@ def validate_cum_func_with_skipna(skipna, args, kwargs, name):
 LOGICAL_FUNC_DEFAULTS = dict(out=None, keepdims=False)
 validate_logical_func = CompatValidator(LOGICAL_FUNC_DEFAULTS, method="kwargs")
 
-MINMAX_DEFAULTS = dict(out=None, keepdims=False)
+MINMAX_DEFAULTS = dict(axis=None, out=None, keepdims=False)
 validate_min = CompatValidator(
     MINMAX_DEFAULTS, fname="min", method="both", max_fname_arg_count=1
 )
diff --git a/pandas/core/arrays/categorical.py b/pandas/core/arrays/categorical.py
index f7e2ebdc8a17a..c7f99866bdfc0 100644
--- a/pandas/core/arrays/categorical.py
+++ b/pandas/core/arrays/categorical.py
@@ -2125,7 +2125,7 @@ def _reduce(self, name, axis=0, **kwargs):
         return func(**kwargs)
 
     @deprecate_kwarg(old_arg_name="numeric_only", new_arg_name="skipna")
-    def min(self, skipna=True):
+    def min(self, skipna=True, **kwargs):
         """
         The minimum value of the object.
 
@@ -2144,6 +2144,7 @@ def min(self, skipna=True):
         -------
         min : the minimum of this `Categorical`
         """
+        nv.validate_min((), kwargs)
         self.check_for_ordered("min")
 
         if not len(self._codes):
@@ -2160,7 +2161,7 @@ def min(self, skipna=True):
         return self.categories[pointer]
 
     @deprecate_kwarg(old_arg_name="numeric_only", new_arg_name="skipna")
-    def max(self, skipna=True):
+    def max(self, skipna=True, **kwargs):
         """
         The maximum value of the object.
 
@@ -2179,6 +2180,7 @@ def max(self, skipna=True):
         -------
         max : the maximum of this `Categorical`
         """
+        nv.validate_max((), kwargs)
         self.check_for_ordered("max")
 
         if not len(self._codes):
diff --git a/pandas/tests/arrays/categorical/test_analytics.py b/pandas/tests/arrays/categorical/test_analytics.py
index 6af3d18fa24ff..1fca0fa5c3f2b 100644
--- a/pandas/tests/arrays/categorical/test_analytics.py
+++ b/pandas/tests/arrays/categorical/test_analytics.py
@@ -1,3 +1,4 @@
+import re
 import sys
 
 import numpy as np
@@ -105,6 +106,37 @@ def test_deprecate_numeric_only_min_max(self, method):
         with tm.assert_produces_warning(expected_warning=FutureWarning):
             getattr(cat, method)(numeric_only=True)
 
+    @pytest.mark.parametrize("method", ["min", "max"])
+    def test_numpy_min_max_raises(self, method):
+        cat = Categorical(["a", "b", "c", "b"], ordered=False)
+        msg = (
+            f"Categorical is not ordered for operation {method}\n"
+            "you can use .as_ordered() to change the Categorical to an ordered one"
+        )
+        method = getattr(np, method)
+        with pytest.raises(TypeError, match=re.escape(msg)):
+            method(cat)
+
+    @pytest.mark.parametrize("kwarg", ["axis", "out", "keepdims"])
+    @pytest.mark.parametrize("method", ["min", "max"])
+    def test_numpy_min_max_unsupported_kwargs_raises(self, method, kwarg):
+        cat = Categorical(["a", "b", "c", "b"], ordered=True)
+        msg = (
+            f"the '{kwarg}' parameter is not supported in the pandas implementation "
+            f"of {method}"
+        )
+        kwargs = {kwarg: 42}
+        method = getattr(np, method)
+        with pytest.raises(ValueError, match=msg):
+            method(cat, **kwargs)
+
+    @pytest.mark.parametrize("method, expected", [("min", "a"), ("max", "c")])
+    def test_numpy_min_max_axis_equals_none(self, method, expected):
+        cat = Categorical(["a", "b", "c", "b"], ordered=True)
+        method = getattr(np, method)
+        result = method(cat, axis=None)
+        assert result == expected
+
     @pytest.mark.parametrize(
         "values,categories,exp_mode",
         [