Moving a pair of studs 4 inches
If you aren't changing the framing of the former exterior wall at all, you can basically do whatever is most convenient for the "wall" on the addition side.  Flat studded walls are not going to be load bearing, so you don't need a header.  The 2x4s on that side are basically only there to provide a place for the sheerock to attach.  Basically you'll be putting in 2x4 furring strips around the hole.  About the only thing that you should try to do is to attach whatever you put in to something solid (i.e. other framing) so the drywall around the window doesn't crack.