Previous versions of the map

Use overpass turbo (https://overpass-turbo.eu/). Create a bounding box over your desired area, then use the following code and/or use the examples and resources on the webpage to tailor the code to isolate the data that you want (the below code loads all drinking water objects as they existed on 2016-01-01). Export the data to kml and upload to qgis!

[date:"2016-01-01T00:00:00Z"]; ( node [amenity=drinking_water] ({{bbox}}); way [amenity=drinking_water] ({{bbox}}); relation [amenity=drinking_water] ({{bbox}}); ); out body;

; out skel qt;

Previous versions of the data, yes; previous versions of the map, no.

This means you can get the data but making it into a map is your own responsibility. For getting the data for a specific point in time,

1. download "history planet" file from https://planet.openstreetmap.org/pbf/full-history/ or alternatively download a regional history file from download.geofabrik.de (that server has history files for individual countries but you need to log in with your OSM account to access them)
2. use the osmium command line tool (see https://docs.osmcode.org/osmium/latest/osmium-time-filter.html) to extract the data for one specific point in time from the history file

If you then want to make a map from the resulting file, you'd basically follow standard instructions for a tile server setup e.g. switch2osm.org.

If you want to generate a map that looks exactly like the map on OSM looked at the time, you will want to check out a suitable older version of the openstreetmap-carto map style.