TCarlos Ghosn, the former Nissan's chairman, fled Japan to Lebanon on the last day of 2019.

Or: Carlos Ghosn, Nissan's former chairman, ...

Typically, essays or newspaper articles will mention names referenced in the title again the first time they are used in the body of the article.

He was first reported to have left Japan hiding in a musical instrument case, which turned out to abe just a rumor as the security footage showed his leaving home alone.

Maybe it's just me but, it took me a while to understand that it was an instrument case, and I was actually thinking about 'case' like "murder case" or "police case". (^^;)

The former Nissan's chairman fled Japan to Lebanon on the last day of 2019.

In English, most company names do not end in possessives. There are a few that do (ex. "Standard and Poor's" or "Sotheby's") but it is not common.

He was first reported to have left Japan hiding in a musical case, which turned out to abe just a rumor as the security footage showed his leaving home alone.

You have to say "turned out to be" in this phrase, saying "turned out to" by itself does not make sense. The article "a" goes after "rumor," not after "just". When you have it as "a just rumor" you are saying "a rumor that is just" (using the meaning of "just" as fair, deserved, which doesn't make sense here). You need to say "just a rumor" to use the meaning that you are trying to use here.

He reportedly used a private jet and an alias to leave the country.

What you had is grammatically correct, but its sounds better to me to explicitly say what he used them for.

More information is awaited regarding how he exactly he departed from Japan.

SomeA French newscaster in France commented on sayingthat it was strange that Japanese people did not appear sovery irritated about this escape.

Saying "some <noun>" to refer to a specific person/company sounds very informal and if you know the identity of the person, it comes off as dismissive of them. If you didn't know and couldn't find out, you would say "a/an" instead of "some." If you are writing a report/article yourself, it would be best to include the actual name or at least their news service (ex. A newscaster with the AFP..."). After the phrase "commented on" we would put the topic that they are commenting about (the escape). So you could say "newscaster commented on the escape, saying that it was..."). But since it is obvious what the comment is about, I would just leave it out. You could just say "newscaster said it was" but "newscaster commented" sounds a little nicer.

He wasGhosn had been charged with financial misconduct.

I think it would be best to use his name here. If you use the pronoun "he" it initially sounds like you might be talking about the newscaster. Based on the rest of the sentence, readers could figure it out, but it is not a very smooth transition. Using "was" is OK, but "had been" emphasizes that we are talking about something that happened well before the events of this story.

In my opinion, if it had beenthe charges were related to tax money that the public paid, they would probably have paid more attention and become exasperated.

This sentences is basically fine but using "it" comes off as kind of informal here. At least in America, the phrase "tax dollars" or "taxpayers dollars" can be used to express what you are trying to say here.

The former Nissan's chairman fled Japan to Lebanon on the last day of 2019.

He was first reported to have left Japan hiding in a musical case, which turned out to ato be just a rumor as the security footage showed his leaving home alone.

More information is awaited regarding how he exactly he departed from Japan.

Some newscaster in France commented on saying it was strange that Japanese people did not appear soto be irritated about this escape.