This app was mentioned in 5 comments, with an average of 2.20 upvotes
The alerts are through Japanese phone carriers. You can download apps that will also receive alerts such as Yahoo disaster alerts, But it is in Japanese. Maybe try looking on the US or whatever country's embassy in Japan website is and perhaps they have a listing for ones in English or some other language.
https://play.google.com/store/apps/details?id=jp.co.yahoo.android.emg&hl=en
dマーケット is an app made by Docomo for shopping for games, apps, and other things. Although now that I'm looking at it more closely, it looks like it just redirects you to the Play Store in the end anyway (I've never actually used it to buy or install anything).
That's unfortunate that you can't use Area Mail.
I've seen others mention this Yahoo app as a possible alternative to a native disaster alert app, but other people in this thread are saying that they got the alert from it late today.
Yahoo Japan has a warning app but it's in Japanese. Woke me up at 7am with a North Korean missile alert one time.
https://play.google.com/store/apps/details?id=jp.co.yahoo.android.emg
Y! 防災速報
https://play.google.com/store/apps/details?id=jp.co.yahoo.android.emg&hl=ja
I found out the app just opens up the map in yoru mobile browser so here's the direct link
https://crisis.yahoo.co.jp/map/
This is the yahoo one: https://play.google.com/store/apps/details?id=jp.co.yahoo.android.emg
There's no English though and you'll get a lot of unnecessary alerts unless you change the settings.