I did a search and found this review of several plugins that MIGHT do the job
You might be able to configure one of them to show just a single post, then you just post who is on duty today, and it gets displayed. You could create a new user, and post the current person on duty under that username, and configure to show only that user’s most recent post so that other unrelated posts don’t appear. Obviously, start with the free ones
Apart from these ones, I think that the plugin ContentViews - https://en-gb.wordpress.org/plugins/content-views-query-and-display-post-page/ - could do that but you might have to go for the paid version (although there is a free version). Apparently, the authors are very responsive to enquiries.
I didn’t read this review too closely, but there are another set of plugins here that might help you https://www.elegantthemes.com/blog/tips-tricks/wordpress-category-plugins Here you could set a special category of posts - onduty posts - and try something similar as well.