I have added my iCloud email address to my Gmail app on my Android phone by generating an application (App) specific password on the Apple website and using the generated password to log in. You should be able to login to Apple using your me.com address as your Apple ID.
www.howtogeek.com/442568/how-to-set-up-icloud-email-access-on-android/
You should be able to access your me.com mails via the Internet too on your Android and put a short cut on your Home screen.
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