As to sewing it on…. get a needle and some thread that matches the color of the jacket or button or both (your choice).
Thread the needle (if you go to Walgreens, you can pick up a cheap emergency sewing kit—it will have a weird looking metal and wire device. Shove the wire diamond through the eye of the needle, insert thread, and pull back—voila!).
Figure out how loosely or tightly the button should be attached to the jacket by looking at the other buttons. Find something that thickness and use it as something solid between the button and jacket.
Knot the thread. Sew up through the jacket first so that the knot is caught on the inside. Sew into one hole on the button, turn the needle upside down and sew right down through another hole. Continue this process a few times.
When you are done, cut the thread and knot it on the underside of the jacket. This means your last stitch is down, through the button and jacket so that the needle ends up on underside.
Yes, I just wrote out fairly common sense instructions, but with no idea how foreign sewing is to you, you might need them.