Is there a way for me to specify exactly when "display weekly" means? E.g. I want the popup to show up at 9am Wednesday, or something like that.
There is not a way specify times (or days) in the Software & Updates control panel.
Generally, it shouldn't be necessary for most folks: The displayed packages don't include time-sensitive security patches (your settings are to install those automatically right away). The only upgrades displayed will be non-time-sensitive bugfixes and backports.
The actual Software Updater application is available from the terminal at any time:
If you really want
update-manageron a timer, you can use a cron job or a systemd timer. However, trying to run the application (which requires a $DISPLAY variable) that way might be more complex than is worthwhile for some users.
apt-daily.timer. There are time fields in the service files that may affect the scheduling you're seeing in the GUI.