Look for a seamstress or shoe maker in your area. Assuming the patch is fabric it is best to sew it on. A sewn patch will lie flatter and be less prone to peel off.
You have an inner liner and the outer layer of your jacket. Find a place where you can separate the inner liner from the outer layer. Sometimes it is at the bottom of the jacket, sometimes you can see it on the inside liner of a sleeve, sometimes it is along the zipper, there are many methods used, so look carefully. This will give you access within the jacket to sew the patch on to only the outer layer. You want the inner liner to be free of the patch as when you don and doff the jacket the liner should be free to move and allow you to slide into the jacket. If you have any holes in your pockets or any other rips in the outer layer you can do these repairs as well, from the inside.
Yes, it is easier and cheaper to sew through the two layers, but don't do it. The inner liner is meant to move independently.
Attach the patch to the outer layer using some spray on adhesive, just to keep it in place. You could pin it, but the pins might move. Using upholstery thread sew the patch to the outer layer only. You could try to use a home machine. It will be awkward but with care and patience it can be done. A patcher sewing machine, mean for sewing shoes, has a long arm and will be easier. You shove the long arm through the hole and start sewing.
The last step is to resew the hole between the inner liner and the outer layer. If you do the job properly then you'll not need to do it again in the future!
