Determining CRS from given point coordinate set
Also you can try using projfinder. For instance, when one paste provided coordinates (4581211.88, 5811848.94)
You will see several options. Then you may assume the CRS, which is mostlikely the "EPSG:31468 | DHDN / 3-degree Gauss-Kruger zone 4".
But in your case I will simply follow what @Erik explained.
And then answering your another question "How can I transform it to EPSG:25832?" It depends on which software/libraries you have access to. It seems to be a new separate question for GIS SE. Maybe you can try MyGeodata Converter.
References:
- Defining and changing unknown CRS into WGS84
It says right there in your column title "GK4" - which is for Gauss-Krüger Zone 4 (https://epsg.io/31468).
Another hint is the leading 4 on the X-coordinates.
An last but not least you should know where the data should be situated and thus reduce the CRS-options.
You can't determine the projection of a set of points just from the coordinates. Without some other information they are just random numbers. Go back to the data supplier and make them give you some metadata.
If you know where the points are supposed to be then it may be possible to guess which projection they are in.