Store the assertRaises() context manager, it has an exception attribute for you to introspect the exception raised:
with self.assertRaises(ValidationError) as exception_cm:
validate_zipfile(test_zip_path + '.zip')
exception = exception_cm.exception
self.assertIn('Zip file not in correct format.', exception.messages)
You could use the Django-specific assertRaisesMessage() method, but take into account that that test does a simple substring test (e.g. you could potentially run into a false positive where you test for a substring of a longer message). Since ValidationError handles a list of messages, a test against ValidationError.messages is going to be more robust.