Seems like there are two ways to solve this problem.

1. Have something outside your network try to contact something within your network. This means you need to expose some device so it can be contacted from the outside. First step is to enable Dynamic DNS, so that whatever solution you go with can keep up when your IP address changes. From there, it could be as simple as enabling the Respond ICMP Echo (ping) Request from WAN setting in the Firewall controls for your AC-87U. Then have a cloud monitoring service try and ping your Dynamic DNS domain. If it gets a response, your WAN is up. If it fails, then alert. Plenty of variations on this, including exposing devices behind your firewall, but the concept is the same.

2. Have something within your network test the connection out. The most reliable solution is to run some sort of monitoring software on a computer on your network. (PRTG is a product I've used in the past and is free for small uses.) Only issue is, when the monitoring software finds the WAN is down, it still needs a way to alert you. If it can't access the internet, how's it going to do so? Your RT-AC87U supports Dual WAN, so you can always use a cellular data connection as a backup WAN, which would allow your monitoring solution to alert you if the primary WAN connection gets disrupted. One downside to this option is that it's possible your entire monitoring solution may get taken down if, say, there's a power outage. So you'd need your computer on a UPS battery backup that will keep it alive long enough to send the alert.

#1 definitely seems easier.