First, you have to load the data into QGIS, and tell it that the data should be treated as EPSG:31467. Normally you do that with right-click ->
Set CRS for layer.
Then, save the data as shapefile in EPSG:31467, and add the file to the canvas. Check if QGIS has chosen the right CRS: The new layer should align with the old, and rightclick ->
Set CRS should point to EPSG:31467.
In step three, you can Save As in another CRS.
By the way, your method worked for me without problem. Did you get an error message saying that
.shp is not a valid layer? Then you forgot to choose a filename for the export. It might help setting the project CRS to 31467 too. Other chance is that QGIS has added the new shapefile with the wrong CRS.
EDIT Here is my simple dataset:
Nr E N
1 3324000 5690000
2 3338000 5698000
3 3330000 5686000
If you want the new coordinates to be in the attribute table, use the field calculator with $x and $y. Make sure to choose fields of type real, not integer; and precision greater 5.
Copying the attribute table into clipboard looks like this:
wkt_geom Nr E N x neu y neu
POINT(6.474297 51.317527) 1 3324000 5690000 6.4743 51.31753
POINT(6.671295 51.393530) 2 3338000 5698000 6.6713 51.39353
POINT(6.562184 51.283429) 3 3330000 5686000 6.56218 51.28343