I don't know where you got these sentences from, but the second sentence is completely wrong. It is grammatically incorrect and it does not really make sense as a sentence either, even if the grammar were to be corrected. Fragment one and sentence four are somewhat unusual, I would say, although not 'wrong.' The answer to your question is no. In one, three and four the word is used in the sense of 'old fashioned.' Completely anachronistically - in a totally old-fashioned way. It appears to be unfortunately and anachronistically necessary - something needs to be done because it has traditionally been done that way. Let's look at this phenomenon anachronistically - let's look at this phenomenon using a very old-fashioned way of thinking about it

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. "Completely anachronistically." 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.

The word has only one meaning, as far as I know, and that 's the one that you have written in the first sentence. The second example sentence doesn't make any sense to me. As for the other two, it's difficult to explain the meaning without any context. The word is more commonly used in its noun form, anachronism.

None of those sentences work. Anachronistic means "in the wrong time." So, for example - "The movie took place in the 1600s, but all of the characters were anachronistically using cell phones."