Can I run rank updates on demand, and if so - how?

The functionality is available starting from Pro plans.

"Update On Demand" can be executed via the URL View "Bulk Actions" menu (top left). Check the URLs you want to run an update for, and click "Update On Demand". All of the terms under these URLs will be updated.

To update selected terms of one URL in the URL View, you can select the terms and click the update icon at the bottom of the table.