Using Classic Outlook on Windows for personal email, calendar, and contact management
If you're using the desktop version of the New Outlook, you'll need to click the Open in New Window button (see red arrow), and then you will have the option to right-click to Save As. CTRL+S will also allow you to Save As the document without having to download it.
If you're using Outlook in the browser, you can just Preview the attachment and RIGHT-CLICK to get the SAVE AS option in the drop-down menu.