It doesn't sound legitimate at all.
It sounds like a phishing email.
You can log into your hotmail account yourself, and make sure things are working, but don't click the link in that email.
You can report the phish if you want.
There is information on how to report a phishing email for Outlook and Hotmail
***here***.
For non- Outlook or Hotmail customers should address an email to report_spam@outlook.com. Attach a copy of the spam email to this email and send it. If we do not receive a copy of the email as an attachment our ability to take action against the spamming account is severely impacted.
Note: We ask for a full copy of the email so that we have conclusive evidence showing from which Outlook/Hotmail account the SPAM was sent. It is possible that the email address was spoofed, and we don't want to mistakenly close a non-spamming account.