Yes. The Egyptians are considered the first archaeologists when they discovered the Sphinx 500 years after it was originally (~2500 BC) built. King Nobonidus of Babylon has the first "muesum" where he showcased earlier Babylonian artefacts. Also upper class Romans were fans of collecting Greek artwork.

In Greece and Rome it was common to note significant battles by displaying souvenirs like enemy shields (the Temple of Apollo in Delphi was decorated with Persian shields) and the bows of ships (like the rostra on the Roman Forum).

Temples also housed famous art. Pliny the Younger mentioned donating a statue to a local temple with a "modest" inscription about himself.

Finally there was a lot of history in public space. The Roman Forum was full of monuments and statues honoring events and heroes from the past. Augustus's Forum was surrounded by a portico filled with statues of famous Romans. Trajan's Column (and it's imitators) had a spiral inscription giving a visual narrative of his Dacian campaigns.

In Rome the most significant form of historic preservation were families. Great families would display busts of their ancestors and keep track of the family history which would be recited at funerals. Successful generals and governors also took items from foreign countries to bring home and display. The most infamous example was Verres the governor of Sicily who seized a lot of art and gave it to Senators as a bribe. Cicero mentions seeing the art displayed in Verres's home and the homes of his friends. So elite homes probably had a lot of material on display from their family past.

None of these is a proper museum, but I think they fit part of the spirit of preserving the past through materials and displaying them to the public.