You could definitely ping your static IP to see if your modem is online, and you could set-up an alert if the ping test fails, but if you're using a UPS for your modem then that won't help much, since the power could be out but your modem could still be online. You'd probably want to check on the status of a device that doesn't have a backup power source.
Since there's no way to externally set-up a ping to a device on your network (unless you're looking to expose a device past your NAT), then unless you have a device like a raspberry pi on hand, the most straightforward, cheap approach might be to purchase a smart WiFi-connected plug / power strip, since those have functionality to alert you if the device loses power.
You could definitely ping your static IP to see if your modem is online, and you could set-up an alert if the ping test fails, but if you're using a UPS for your modem then that won't help much, since the power could be out but your modem could still be online. You'd probably want to check on the status of a device that doesn't have a backup power source.
Since there's no way to externally set-up a ping to a device on your network (unless you're looking to expose a device past your NAT), then unless you have a device like a raspberry pi on hand, the most straightforward, cheap approach might be to purchase a smart WiFi-connected plug / power strip, since those have functionality to alert you if the device loses power.