The free Outlook apps don’t have a built-in “text me when this sender emails” feature, so you have to create a workaround. A couple of available options include forwarding important emails to your phone’s SMS gateway (which turns emails into texts), or using iPhone automations to convert certain email notifications into messages. The SMS gateway method is more reliable because it works across your iPhone and Mac without needing apps running.
Set up a rule in Outlook.com (web version, not the app) that looks for emails from specific senders and automatically forwards them to your phone’s SMS email address. Go to Outlook.com in a browser, open Settings, then Mail, then Rules, create a new rule with a condition like “From = ******@sender.com” and an action “Forward to” your SMS gateway address. Your carrier converts that forwarded email into a text message on your phone. You can find your gateway format below; replace the number with your own phone number.
Verizon: ******@vtext.com
AT&T: ******@txt.att.net
T-Mobile: ******@tmomail.net
Google Fi: ******@msg.fi.google.com
US Cellular: ******@email.uscc.net
If you prefer to keep everything on your iPhone instead of using carrier gateways, you can build an automation in the Shortcuts app that watches for Outlook notifications and sends you a text to yourself or another number, but this is probably less reliable because AFAIK iOS does not allow Outlook email triggers directly the way it does for Apple Mail.
If the above response helps answer your question, remember to "Accept Answer" so that others in the community facing similar issues can easily find the solution. Your contribution is highly appreciated.
hth
Marcin