Were ancient people interested in archaeology?

St Augustine, in the City of God, citing Livy

A certain one Terentius had a field at the Janiculum, and once, when his ploughman was passing the plough near to the tomb of Numa Pompilius, he turned up from the ground the books of Numa, in which were written the causes of the sacred institutions; which books he carried to the prætor, who, having read the beginnings of them, referred to the senate what seemed to be a matter of so much importance. And when the chief senators had read certain of the causes why this or that rite was instituted, the senate assented to the dead Numa, and the conscript fathers, as though concerned for the interests of religion, ordered the prætor to burn the books."

So, the Roman Senate was quite interested in books found near the tomb of a famous ancient king, supposed to be the first successor to Romulus, and credited with originally setting much of their pagan rites as St. Augustine explains elsewhere and wiki confirms. But when the contents were not of their liking ("dangerous to religion", as Livy says), they burned them.

The Talmud records an (apparently failed) archeological attempt at establishing a particular point of rabbinic law:

Rabbah b. Bar Hana related: We were once travelling in a desert and there joined us an Arab merchant who, [by] taking up sand and smelling it [could] tell which was the way to one place and which was the way to another. We said unto him: 'How far are we from water?' He replied: 'Give me [some] sand.' We gave him, and he said unto us: 'Eight parasangs.' When we gave him again [later], he told us that we were three parasangs off. I changed it; but was unable [to nonplus] him.

He said unto me: 'Come and I will show you the Dead of the Wilderness.' I went [with him] and saw them; and they looked as if in a state of exhilaration. They slept on their backs; and the knee of one of them was raised, and the Arab merchant passed under the knee, riding on a camel with spear erect, and did not touch it. I cut off one corner of the purple-blue shawl of one of them; and we could not move away. He said unto me: '[If] you have, peradventure, taken something from them, return it; for we have a tradition that he who takes anything from them cannot move away.' I went and returned it; and then we were able to move away. When I came before the Rabbis they said unto me: Every Abba is an ass and every Bar Bar Hana is a fool. For what purpose did you do that? Was it in order to ascertain whether [the Law] is in accordance with the [decision of] Beth Shammai or Beth Hillel? You should have counted the threads and counted the joints.

^{(Babylonian Talmud, Bava Batra 73b, Soncino translation)}

The protagonist, Rabba Bar Bar Hana, who flourished some 1,700 years ago was dealing with artifacts from well over 1,000 years prior to that.

In the biblical book of 2nd Kings chapter 22, there's the story of Josiah repairing the temple and his high priest discovering an old scroll of the Torah. They have it read out to the court, realize how far they've strayed, and tear their robes and recommit.

[Josiah to servants] “Have these men pay the workers who repair the temple of the Lord — the carpenters, the builders and the masons. Also have them purchase timber and dressed stone to repair the temple” ... Hilkiah the high priest said to Shaphan the secretary, “I have found the Book of the Law in the temple of the Lord.” ...

Shaphan read from it in the presence of the king. When the king heard the words of the Book of the Law, he tore his robes. He gave these orders ...: “Go and inquire of the Lord for me and for the people and for all Judah about what is written in this book that has been found. Great is the Lord’s anger that burns against us because those who have gone before us have not obeyed the words of this book.”

Whether you see the story as history from the 7th century BC or a retrospective account written around the 1st century BC, it's a pretty old reference to archaeology. Not scientific, carefully planned archaeology, of course — but clearly they were familiar with the experience of rediscovering remnants of older strata of civilization.

This depends on definition.

Surely people who were interested in buried treasure or religious relics existed all the time. But they mostly were looking for valuables for personal gain or religious insight than were motivated by scientific interest.

During the time of Jesus, for instance, there was one would-be-prophet who was leading people around claiming he knew where some ancient holy vessels were buried.

In his commentary on Beowulf, Tolkien makes the point that the dragon's hoard of gold in his barrow represents an ancient view of archaeology, of an attempt to explain remnants of older civilization:

The scene in the barrow passes at once into an elegiac retrospect on the forgotten lords who placed their gold in the hoard, and then died one by one until it was left masterless, an open prey to the dragon ... The feeling for the treasure itself, and this sense of sad history, is just what raises the whole thing right above 'a mere treasure story, just another dragon-tale'. The whole thing is sombre, tragic, sinister, curiously real. The 'treasure' is not just some lucky wealth ... It is laden with history, leading back into the dark heathen ages beyond the memory of song, but not beyond the reach of imagination. ...

So this passage rivals the exordium on ship-burial as that very rare thing, an actual poetic expression of feeling and imagination about 'archaeological' material from an archaeological or sub-archaeological period. Many such mounds existed in Scandinavia, and even in England in the eighth century, already ancient enough for their purpose and history to be shrouded in mist. Here we learn what men of the twilight time thought of them. ... the ashes of Beowulf himself are now to be laid in a barrow with much of this same gold and pass down into the oblivion of the ages.

As I remember, the Babylonian sequence of the movie Intolerance (1916) describes Prince Belshazzar in charge at Babylon while King Nabonidus is away doing archeological excavations. I think that the title cards used the word archaeology or archaeological.

From Wikipedia, Nabonidus (who reigned from 556–539 BC):

Nabonidus is known as the first archaeologist. Not only did he lead the first excavations which were to find the foundation deposits of the temples of Šamaš the sun god, the warrior goddess Anunitu (both located in Sippar), and the sanctuary that Naram-Sin built to the moon god, located in Harran, but he also had them restored to their former glory. He was also the first to date an archaeological artifact in his attempt to date Naram-Sin's temple during his search for it. Even though his estimate was inaccurate by about 1,500 years, it was still a very good one considering the lack of accurate dating technology at the time.

I am sure there are experts on the history of archaeology who have opinions on how much Nabonodus's activities should count as "the worlds first archaeological work".

But until someone gives strong evidence that there was archaeology before Nabonidus, or that what Nabondidus didn't wasn't really archaeology, we might as well accept that Nabonidus "initiated the world’s first known archaeological work".

The true cross is supposed to have been found in an excavation in the 4th century CE.

Not so much related to archeology, but historical sightseeing, at least for propaganda purposes, definitely was a thing. E.g. Xerxes, Alexander the Great, Caracalla and Julian the apostate all visited (what they thought to be) the ruins of Troy.