I learned how to set the inset map in the data frame properties.
I would like to export a permanent inset map layer or shapefile when I finish my map.
What is best way to do it or is there any alternative method?
It sounds like you have used a dataframe to configure an inset map in your map document (mxd), probably using an extent indicator, which makes sense.
A dataframe is a container of layers so a method to export one of its vector layers as a shapefile is to go to the Table of Contents and Export > Data.
If you are looking to "re-use" a multi-layer inset map dataframe in another map then your best option may be to group its layers and then on that layer group use Save As Layer File.
You could also try copy/pasting a dataframe (your inset map) from one MXD to another but I cannot recall having tested that so it may not work.