Why didn't Sauron unleash the Nine on the Last Alliance?
 

Surely Sauron's greatest servants could have done significant damage against the heroes of the West. Or were the elves and dunedain too powerful for even them to overcome?

Because they would not be effective against armies of Elves and Dúnedain freshly arrived from Númenór. The chief weapons of the Nazgul were fear and intimidation. Look at it this way, if Gil-Galad and Elendil were not intimidated facing Sauron wearing the One Ring, then the Wraiths would be useless.

Next.

Don't be misled by the movies. PJ could only envision the Nine as physical foes- which in general they were not. Their weapon was sheer overpowering unmanning disabling FEAR (plus some sorcery by the W-K).

And don't underestimate that power. Wargamers and RPGers have an idea of what having nine area-of-effect "Morale -100" units can do. In warfare, morale is everything. It is difficult to define, but it is the result of the alchemy of personal will, patriotism, love of comrades, concern for reputation, fear of punishment, training, tactical situation etc etc etc which, at the basic level, keeps soldier performing the entirely unnatural act of standing his ground and not doing what millions of years of evolution are telling him: "Get out of here! Fly, you fool!" The ultimate goal of a battlefield commander is to break the enemy's morale. Not slaughter them all to a man (that almost never happens), but inflict enough casualties and present enough of a present danger that the other side packs it in and either retreats, or panics and runs, or surrenders. As Napoleon said, "A battle is merely two large groups of men trying to frighten each other."

But, as mentioned, Elves cannot be daunted by the shades of Men; not Legolas in the Paths of the Dead, and certainly not High-Elves of the West (on both known occasions, the Nazgul ran away from Glorfindel.) And Numenoreans fresh from Westernesse, not the dwindled Men of Gondor of some 3 millennia later, might not have been wholly immune but they would have been far more in the position of Earnur at Fornost.

Since Elendil, Isildur, Anarion and other high ranking warrios just came from Numenor, I guess their spirit and strength must be higher than Earnur and will have little trouble facing the Nine?

You might be interested to read this recent discussion on the Nazgul. :)

http://forum.barrowdowns.com/showthread.php?t=19437

Which, I will also take the time to thank the members who responded to that thread and I have more thoughts I've been meaning to get to and respond.