||>on my own way
Oh yes, that's your own way alright.
On the one hand, I can see how that does work to some very coarse degree. But I cannot imagine ever recommending that to someone as a reasonably viable solution. You will lose some degree of data accuracy following such a workflow.
If you have a shapefile that does not have a .prj file, AND you want to convert that shapefile into a different projection or coordinate system, you can use ArcCatalog and its Toolbox for doing so.
In ArcCatalog, right-click the shapefile and select "Properties". Then on the "XY Coordinate System" tab, use the "Select", "Import", or "New" buttons to assign to the shapefile a description of its projection and coordinate system. Good chance that most of the time the "Select" button is fine.
The "Project" tool allows you to create a new output shapefile that is transformed into some different projection and coordinate system.