Since moving from Hotmail to Outlook I can no longer access emails on my Android device. I have installed, removed, and reinstalled the Outlook app, but despite my receiving emails daily my inbox is blank. How do you get to use your Hotmail account on your mobile device?
Usual solution is to remove and re-add the account, on your mobile Outlook.com. Generally that works. (Obviously you will lose any mails and unsent outgoing mails (manually save unsent outgoing mails first with some file-manager app, if you need them)).
Be warned: Outlook.com's Android sync is very brittle, and their support is nearly non-existent. Beware of taking over-the-wire Android updates.