No (o A o), it is pronounced how it is spelled, there is a reason it is spelled like "Hatsune". The romanization of 初音ミク is Hatsune Miku, as that is her original and official name, as shown directly on product covers and so for from Japan . . . not Hatune o A o
The "tsu" phoneme is pronounced "zu","du", or "dzu". It is most commonly translated to either "tsu" or "tu" depending on the translator being used. Just because the hirigana dictates that it is spelled on way does not mean that it is pronounced that exact way. Japanese pronunciation is generally straightforward but there are some exceptions. Just like how "ら", or "ra" isn't pronounced with an english "r" like in "robot", but more of a combination between "r","l", and "d"
Sorry If its too long or if I didn't elaborate in the original reply.