Anachronistic does not mean old-fashioned - it means that something is in a time period where it DOESN'T BELONG.
You could say, for example, that wedding rings are an anachronism today, because they date back to the time when marriage indicated ownership, and so they have no place in the modern world.
But, in fact, it is just as common to use the word to refer to something which is too modern for the supposed period. For example, if you see a play set in the early 1800s and someone uses a telephone, this is anachronistic. Telephones did not exist at that era, so this mistake is an anachronism.
This seems natural. Let's say you're watching a film set in the year 1960. You might say 'Then, completely anachronistically, the two girls started singing a Beatles song.' This is anachronistic, because the Beatles weren't known at that time - they had their first hit in 1962.
"We get anachronistically speak our minds"
This makes no sense at all.
"It appears to be unfortunately and anachronistically necessary"
You'd need more context to explain this. Maybe it means that a writer or filmmaker, for example, is being forced to use 'artistic licence' to include something from the wrong era in their work.
"Let's look at this phenomenon anachronistically"
This is an unusual use of the word.
It could mean that the speaker is suggesting that we look at the phenomenon through the eyes of a person from different era. For example, if we interpret a phenomenon from ancient history according to our 21st-century values or based on our 21st-century knowledge or experience, or if we look at something modern as if we were from an earlier period.
That's possible. But I actually suspect that this speaker is using the word incorrectly.