I am aware that Azure Storage Account has the availability alert. What about Azure App Service? Does it support availability alert too? I am not considering Application Insights because it is still under preview and there is no SLA.
You can setup an amount of alerting from the normal App Service itself, if you go to the portal, select the Web App, API App or Mobile App in the app service, there is a Monitoring section which has an Alerts page on it- this will allow you to configure things like "Let me know when there are more than 5 Http 500 errors in a 10 minute period".
If you'd like availability monitoring however, then this is reported within Application Insights. There are a couple of different types you can run in here, standard "ping" tests against your endpoint that look for a 200 response, or you can run web tests that you've pre-authored/recorded and uploaded. You can run these from various countries all around the world.
As for your comment on App insights being in preview - you're in luck: