Literally just

#!/bin/sh

if checkupdates || yay -Qu; then
    notify-send "Package updates available" "To update, press MOD + SHIFT + U" -i "update-catppuccin-mocha"
fi

mod+shift+u was bound to spawn a terminal window running yay -Syu, obviously change the notification to say whatever you want. The icon is a custom icon, replace it with whatever icon you want for the notification or just remove the icon if you don't want one.

I've since moved to Artix so the test is now just yay -Qu as checkupdates doesn't seem to exist on Artix, but if you're on base Arch and use yay, the above should work. You can also remove the yay if you don't use yay and I think that just checks for updates from official arch repos, not from aur. (yay -Qu should check both but I have both commands in the script just in case)