There was a prince in Egypt, one of the sons of Ramesses II (I believe the eldest), who dedicated his life to studying the old pyramids. Keep in mind that by the time of the Ramesside pharaohs, the Giza pyramids were over a millennium old. So this guy, Khaemwaset, went around and examined them, even sometimes restored the superstructures' crumbling facades. Furthermore interesting is that by this time, pharaohs had stopped building pyramids as such - they had become more of the way private individuals buried themselves - and so he was really breaking new academic ground in studying these long-abandoned structures. He was able to devote his life to archaeology as his father lived for a super long time - Ramesses II outlived the first 12 of his sons.

Sources: his wiki page is good, http://en.wikipedia.org/wiki/Khaemweset, but for more detail I'd recommend Shaw, Oxford Encyclopedia of Ancient Egypt, which is available online if you have a university connection. For the New Kingdom pharaohs' turn away from pyramid building, reading about the funerary practices at the "Town of the Tomb Builders" at Deir el-Medina should give insight, as should any article about just "funerary religion": the Shaw book has both entries.

Yes. In the ancient sources both Herodotus confirms that when the Spartans were trying to beat Tegea in order to solidify control of the Pelopenesse in the mid 6th century they were given an oracle to bring their cultural hero Orestes' bones to the city. Eventually someone dug a well and discovered giant's bones (what we probably think was a giant sloth) and they moved it and had a ceremony in their city. Later the Athenians did the same thing in the Pelopennesian war to find the body of Thesues and dug up a Skeleton with a Bronze Spear. They were digging for their version of finding historical relics, but always assumed it was the correct find because they needed it for a present military event. For instance the reason they dug for the "grave" of Theseus was that someone saw an Eagle land at that spot. They used any archeological find as a propaganda and symbolic event God ordained and didn't examine it as anything other than that.

I am not qualified to answer this, but weren't there a lot of Greeks and Romans that spoke at great length about prior cultures? I don't think they had anymore than hearsay and legend, but people like Strabo, Herodotus, Thucydides, etc. discussed wrote histories of varying accuracy. Some of their accounts of contemporary cultures are still used today (often with a grain of salt) because there is little record of these cultures. The Scythians did not record their history, but the Greeks wrote about what they knew of them.

Whilst plenty of people in this thread are answering about past cultures studying older cultures, it is worth pointing out that this very much was not archaeology, as that has existed only really for 150 years. Ideas like stratigraphy and deep time, crucial for understanding excavations, are relatively recent, and before then it was very difficult to establish where/when an object discovered was from.

I asked this a month ago and there were several excellent replies no one put here yet: http://www.reddit.com/r/AskHistorians/comments/31ftzl/did_any_of_the_more_advanced_ancient/

The first chapter of "History of the Persian Empire" by A.T. Olmstead (pdf) starts by talking about it.

Quotes: "When Cyrus entered Babylon in 539 B.C., the world was old. More significant, the world knew it's antiquity"

"Nabu-Naid, last independent king of the Chaldaeans, rejoiced when he unearthed the foundation record of Naram Sin, unseen for thirty-two hundred years - or so his scholars informed him."

You can read the rest in the pdf I linked.

This isn't archeology, but it's worth mentioning. The use of Roman spolia (reused art, usually statues) was pretty common at the courts of Western European rulers in Late Antiquity (300-640). This was mainly because post-Roman polities, which had their own distinct ethnic and cultural identities, moved into previously Roman locations and decided that it was more convenient to reuse Roman infrastructure than to create their own. This resulted in the widespread emulation of Roman designs, images, and buildings throughout the early Medieval world (640-1000).

On coinage, for example, post-Roman monarchs were depicted in a similar composition and style to Roman emperors on Roman coins. Troves of Roman coins could be found anywhere in Europe or the Mediterranean, and often coins that were centuries old remained used for trade.

Entire cities that had remained standing due to the sophistication and durability of Roman engineering became repurposed and emulated. Basilica were rededicated as churches, and palaces were inhabited by local communities. Reccopolis, a bustling Visigothic metropolis founded in the late-6th century, is full of colossal masonry and pseudo-Roman architecture (lots of arches).

One of the coolest examples of Roman spolia becoming a cultural phenomanon is the 9th century Carolingian bishop Walafrid Strabo's poetic dialogue "De Imagine Tetrici". In the text Strabo vocalizes his theological concerns with reusing a statue of the heretical king Theodoric the Ostrogoth, which had been set up in Charlemagne's capital city and was itself a reused statue of a Roman emperor, to depict the most Christian Frankish emperor. I haven't been able to find a translation online, but if you do please pm me!

I could go on and on about Late Antique art, I'm definitely a bit nuts. I know this doesn't count as archaeology since the Roman statues and buildings usually just remained in relatively public places for millennia, but it is kind of cool how, even though Late Antique and early Medieval societies all had their own different cultures and political agendas, they all considered Roman art and engineering part of their own history and identity.

I would highly recommend the more recent books by professor Chris Wickham (his early scholarship on Lombardic Italy is worth reading if you're into neo-Marxist analyses of early Medieval economies), which give an overview of how Late Antique societies inherited the material past of the Roman Empire.