In order to update your phone number you must first be logged-in to Outlook on Chrome.
Once logged in you will see your profile picture or your Initials on the top right corner of the page
Once you click on your profile or Initials in that circle you will see the following pop-up.
Click VIEW ACCOUNT. This will take you to the following page as shown below.
Under Security Info click UPDATE INFO, and on the following page you will see a list of sign in methods you have added. If you want to change or update your phone number click on the CHANGE option next to Default sign-in method followed by your current phone number.
Once you clicked on CHANGE you will get a small pop up window where you can type in your new phone number. Click CONFIRM once you have typed it in. You will now get a code sent to your new phone number.
A new Pop-up window will appear where you will need to enter the code sent to your new phone number to confirm your identity. And you should be good to go.