Edit -> Preferences -> General -> "When new messages arrive:"
Unchecking the "Show an alert"-Box should do the trick.
First, you'll need a local account. If you already have one, you can go straight to the "Email & app accounts" screen (see further down).
If you don't have a local account, you'll need to create one first (don't worry, this won't reset your settings). Only then can you remove your Microsoft account from your computer, which will get rid of the annoying notification.
Go to Settings -> Accounts.
![Accounts](https://i.stack.imgur.com/KSenU.png)
On the "Your info" page, click "Sign in with a local account instead".
![Your info](https://i.stack.imgur.com/DXIoN.png)
Confirm your current Microsoft account password to continue.
![Microsoft sign-in](https://i.stack.imgur.com/kN1oZ.png)
Choose a username for your local account. You can also specify a password if you want to.
![Create local account](https://i.stack.imgur.com/GTxNc.png)
You will be signed out of your Microsoft account and can now sign in with the local account you just created. Once you have done this, go back to Settings -> Accounts and go to the "Email & app accounts" page.
![Email and app accounts](https://i.stack.imgur.com/KYXfb.png)
You should see your Microsoft account listed under "Accounts used by other apps". Click on it, and select the "Remove" option.
![Remove Microsoft account](https://i.stack.imgur.com/3eTZx.png)
This disassociates your Microsoft account with your PC. You no longer need to enter your Microsoft password to log in and you should also be rid of the annoying notification.
Note: If you need your Microsoft account for any of your apps to work, this solution unfortunately will not work for you.
Best Answer
Here's what worked for me, though it does turn off a little more than just that notification.
After getting the notification, don't click on the cross or remove it.