Certainly not with the rigor of the modern discipline (at its best...), but to some extent yes. You mention ancient Assyria, and specifically in ancient Mesopotamia it was often of interest to latter-day Mesopotamians to dig up the cuneiform tablets left from earlier eras and attempt to read these, decipher what they said, and copy them into the writing of the day. Ancient scholars of more-ancient texts even produced lexical tablets listing then-archaic signs and their then-current equivalents to aid them in comprehending archaic cuneiform texts; there were not just language shifts, e.g., Sumerian to Akkadian, to contend with, but also shifts in the meanings of cuneiform signs and replacement of archaic with newer signs. As you might imagine, these lexical translation aids that ancient scholars produced to help each other decipher even more ancient writing have also been useful to modern scholarship (although the conjectures of ancient translators trying to read texts that were already thousands of years old are not always right either).

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There is also evidence of interest in learning about and restoring monuments and temples of the ancient past, and particularly the restoration of ancient temples could attract the attention of not only scholars but rulers interested in connecting themselves to Mesopotamia's shared history and traditions. You'll likely find the name of Nabonidus, the Neo-Babylonian king who was the last to rule before the Persian conquest, thrown around a bit as "history's first archaeologist," and he was certainly one prominent person who seemed to have proto-archaeological interests.

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However, to my knowledge antiquarian interest isn't so much attested outside of the popular topics of a) deciphering archaic texts or b) royal or religious monuments. Ancient Mesopotamians may have taken an interest in the tablets, temples, and monuments of their predecessors - items that bound them into a shared cultural tradition and gave them a strong sense of the great antiquity of that tradition - but it is not clear that you could find them taking the same interest in the tools, rubbish heaps, pot sherds, and other stock and trade items of modern archaeology. So they may have been trying to figure out the age of a temple or they may have found an ancient king's name on a brick and tried to determine when he ruled and what he did, but it's not clear that they were also interested in answering archaeological questions like what tools their ancestors of a thousand years ago used in their daily lives.