You can view the staged version of a file using:
git cat-file -p :./FILENAME
The leading :
causes git to read from the index. The ./
gets it to look in
the current directory, this part may be omitted if you specify a path relative
to the top of the repository rather than to your current directory.
You can view a specific subset of lines by piping the output from that to another command to do the desired limiting as for any other file.