Are the buttons on the jacket accurate? Or will you be buying replacements?
You can sew the button holes closed quite easily using a basic zig-zag stitch, and the original button hole should be mainly covered by the buttons.
As for making new button holes on the other side, I personally hate doing button holes on a machine let alone by hand.
One trick that I might be frowned on for suggesting, hehe, is to use superglue. If you've ever spilt superglue on yourself it makes the fabric very stiff. If you can carefully glue in a straight line (you might need to get tricky and line up and glue from the back as well depending on the thickness of the jacket to make sure all the layers glue properly), you could then cut it with a scalpel and have instant buttons.
Just be aware if you did that, the fabric won't flex as easily.
Then if you want to finish it off you can whip stitch the edges.
(P.s. I hate button holes. Did I mention that?