So the first museum was in Ur around the 6th Century BC and was curated by a local Princess. Leonard Wooley who found the site has this to say:
What were we to think? Here were half a dozen diverse objects found lying on an unbroken brick pavement of the sixth century BC, yet the newest of them was seven hundred years older than the pavement and the earliest perhaps sixteen hundred.
He also discovered cylindrical catalogues of each item which he saw as akin to museum identification tags:
. The room was a museum of local antiquities...and in the collection was this clay drum, the earliest museum label known, drawn up a hundred years before and kept, presumably together with the original bricks, as a record of the first scientific excavations at Ur
However, I'd be reluctant to call it archaeology. Archaeology (to me at least) implies methodologies and some sense of formal academia. This didn't really happen until Kulturkriess, a theory started by Kossina a german proto-fascist. Kossina believed the job of archaeology was to track cultures and their lineages, and that material culture (like pottery for example) can show us this. This kind of took off but it wasn't really scientific. Until V. Gordon Childe decided to adapt it. Childe was a Marxist and believed firstly, that material history is the only way to truly understand the past, secondly, that racism was bad, and thirdly, that the scientific method must be employed within archaeology. It's this point I would say archaeology emerged as a distinct entity, before then it was just theories about objects from the past.
Childe, G.V. 1926 The Aryans, Kegan Paul, London
Childe, G.V. 1931 Skara Brae, Kegan Paul, London