I am the Mouth of Sauron.

[The Might]

I doubt it, and I am sure it refers to the rebuiliding of Barad-dur, completed in 2951, which would make him at least about 80 years old, a normal age for someone of Numenorean descent.
Anyway, any idea of him being more powerful then the Wk makes absolutely no sense, since Tolkien clearly states that the 9 were his strongest servants, and that the WK was the strongest of the 9.
I do not think that because he was going to receive great authority after the War was over it meant that he was the strongest. He was only the best qualified servant of Sauron for the job. He was lieutenant of Barad-dur at the time, meaning that he had been dealing with the tower's internal business for quite some time, whereas the Nazgul were specialised in other things such as killing annoying Rangers and bringing back important rings...
Sauron could have used the Nazgul much better for other purposes, and left the administrative and political duties to Mouth of Sauron. He was clearly a diplomat, and he was the ambassador of Mordor at the Black Gate. Who could be more qualified to take control of newly conquered land? Surely not a Nazgul.