Thanks for sharing your question with our network, @YL2!
To second what fellow user, @Mirdas, mentioned it is a great idea to employ the use of an announcement banner to share important information like this with your customers. You may also consider adding a Shipping FAQ or page for easy access with information about domestic or international shipments and expectations.
Another unique solution is to consider adding a short message next to the name of your rate under Settings > Shipping. That means when you name your rate, for example "standard shipping" you could update that to read "standard shipping [cut off time 1:00AM _insert timezone_].
I trust that helps, but let me know if I missed anything.
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