I've been seeing that ad at the top of my inbox lately also, and I just ignore it. Microsoft is not trying to force you to do anything; you are free just to ignore the ad and not update your account. I have no problems accessing my Outlook email, but you seem to, and I have no idea why. I don't believe the ad and your Outlook access problems are related.

Are you using an EOL Chromebook, by any chance?

Can you give us more info on what your problem is accessing Outlook?