As far as I know, we have absolutely no idea how old the Mouth of Sauron is or even certain knowledge of where he comes from. The best guess is that he is originated in one of the lands conquered and colonized by the Numenoreans, the most famous of which is Umbar. But we can't be certain. He could come from some unmentioned place in the interior, or from somewhere further down the coast.
But it is HIGHLY unlikely that he is 3000 years old, and indeed I would say impossible. The ability to prolong life in that fashion would require a great ring, and it is possible that even Sauron could not fashion another great ring without the assistance of the elves. Certainly thier is no indication that the MoS is associated with a ring, or that Sauron continued to fashion rings, and there is certainly no indication that MoS has faded as a mortal should whose life has been so unnaturally prolonged.
It is actually rather unlikely that he is as much as 300 years old, though that is certainly closer to the ball park. We know that the Numenoreans forfeit to a greater or lesser degree thier right to prolonged life when they act in an univirtuous manner. We know that the Black Numenoreans, that is the racist Numenorean conquerers of the southern reaches who abandoned the worship of Illuvator and became worshipers of Morgoth, have been VERY unvirtuous for generations. So it is highly likely that the Black Numenoreans are greatly decayed and don't live nearly as long as Numenoreans in periods in which 'the blood ran true'. Certainly there is every reason to assume that the MoS is younger than Aragorn was at his death, probably by a century or more.
As a general philosophical theme, I think JRRT would have rejected allowing evil to create life or prolong life. Unless you want MoS to be a wraith, or undead, and I believe the text specifically rejects that he is either, you have to accept a fairly young age for the MoS.
Now, if you are looking for Epic levels of power - you can turn to Sauron, Tom B., the Istari, the Balrog, the Witch King, Glorfindel, and Galadriel. Galadriel is after all in the neighborhood of 30,000 years old, and the daughter of Finwe (so she is probably one of the first elves born of elven flesh on the planet). Of the non-'outsider' characters in LotR she is certainly by far the most powerful, and could probably have at least contended with the Balrog or one of the Istari herself. After all, it took more than one Balrog to kill her brother Feanor, and Feanor once cowed Morgoth, and another one Fingolfin once wounded Morgoth so severely that the pain never left him. Morgoth was Sauron's boss. She is one of the few characters in the story who probably _could_ have taken the ring and defeated Sauron with it. I don't personally feel Aragorn could have, and it is highly likely that if Aragorn claimed the ring and Sauron came up and said 'Give me what is mine.', that Aragorn would be compelled to do so.
The highest level fighter in the story is actually a minor character - Glorfindel. We know that Glorfindel is the very same Glorfindel that killed a Balrog during the fall of Gondolin, so at least in physical might he is the equal of Gandalf.