Why the word order in this sentence is not the usual order? is "begeistert" here the perfect for the verb haben or it is an adjective?
Although this sentence is well-formed German, you are right in supposing that there is something special with the word order. More usual (or non-specific) word order would be:
Mich persönlich hat Wien am meisten begeistert.
Wien hat mich persönlich am meisten begeistert.
Am meisten hat mich persönlich Wien begeistert.
Am meisten begeistert hat mich persönlich Wien.
is a totally acceptable, good sentence. Using this word order you put special emphasis on "Wien", as for example in a context such as
Wir waren in Paris, Florenz, Wien und Buxtehude. Am meisten begeistert hat mich persönlich Wien.
But you would hardly meet this in written communication. It seems somehow restricted to oral utterances.
You may take the entire series of four sentences as a series with increasing emphasis on "Wien".
This is an unusual (but acceptable) word order in German.
Here, you are putting the conjugated verb, "begeistert" in the first slot by "merging" it with "Am meisten," the adverbial phrase.
The purpose is to free up the last place for emphasis. Which is now on the word, Wien.
This is much more common in spoken, than written German. So the phrase
Am meisten begeistert hat mich persönlich Wien," means: "Was most inspired personally (by) Wien" (as opposed to some other city).