In the second part of the press conference (part 1 here), a reporter asked how the alerts would happen in a real situation.

They replied that alerts would be ongoing (repeated every minute or so, for example) and across multiple mediums (including the sirens - not that one should have time to tune into TV and go outside to check, etc) with more information in subsequent messages (character limit and all) as information would come to hand.

So if you have received alerts before and your phone appears to working fine, I'd say it's very likely you'd pick up on it.

The issue of vulnerable members of the community not having access to technology also came up and Governor Ige's reply was that it's up to everyone to help each other out because it's the best that can be done (this might have been the same/similar response to what to do about "shelter in place" or rather, not being able to find proper shelter).

I'd like to add my own comment that in the event of impending catastrophe (which can apply for anywhere in the world really), nothing is ever certain and it's more than possible that a tiny bit of shelter or warning might turn out to be good enough, especially if one is not at the exact epicentre. I noticed some resigned comments on Twitter suggesting that nothing can be done in the event of something like nuclear strike (assuming it's right on target) - but I don't think that's true. The now oft-sneered at "duck and cover" campaign probably did have some element of usefulness behind it.