I would say "Hardly had he heard of the news" means "*almost* as soon as he heard the news."

I would say "just as he heard the news" to mean "as soon as he heard the news" with no "almost."

The former has a sense of cause-and-effect...hearing the news makes him cry.

"Just as he heard the news the volcano erupted" doesn't imply any direct connection between the

events and allows for perfect simultaneity.

In our daily life or in our daily lives, which one is correct?

Do consider "in daily life," no "our" if you want to describe ordinary practices without referring to specific people.

"In daily life, people have hardly any use for radio telescopes."

"In our daily lives, my family members often provide assistance to the poor."

"Hardly had he heard of the news"  -means as soon as he heard the news