If you have flat feet then proper running shoes are THE MUST, without them you are doomed to fail. I will not elaborate too much on the shoes, please look at my dedicated article. But without the proper shoes nothing else will be enough to prevent the horrible pain caused by the fallen arches. On the other hand, even the best shoes just by themselves are not enough for people suffering from flat feet to keep running. Please read further.
Get good insoles into your running shoes. This is very important. Running shoes are made with more or less support, but they never have enough support for people with flat feet. Usually the shoes come with some sort of insoles which can provide extra support, but for hopeless cases like me it has been never enough. Insoles or inserts are the necessary complement of the running shoe. The best is to get the insoles in the same store where you get your shoes at the same time you actually buy your shoes, so you can try how they fit. Usually the insoles for flat feet have more support and thus elevate your foot a bit higher then the original inserts provided by the shoe manufacturer. Therefore not all insoles or inserts may feel equally comfortable in all shoes and you will have to find the best combination. The insoles have to fit exactly, otherwise they can cause blisters by rubbing your foot with the edges of the insole, or just by the insole sliding back and forth in the shoe during running. Again as with buying the running shoes, an expert advice by a qualified personnel can make the difference. Good insoles cost about 30$. They last pretty much as long as the running shoe itself, and for people like me it is definitely a great investment.
Now you have the shoes and the insoles. You should get out and run. And boy, it will hurt !!! Yes, first 3 or 4 runs will hurt quite a lot. Your arches will be in flames, and you will have to gather all your mental strength to overcome the initial pain. But it is definitely worth it. It is just the sign that your fallen arches are forced to get up and adapt to a good support provided by your new shoe and the insoles. It is actually a good pain and it won't last long, I promise. Don't plan on any speed exercise or power run, just go for easy slow jog and concentrate just on your foot steps.
When your run is over you need to stretch a bit like after every other run. But you, as a runner with flat feet, should also stretch your arches, especially if they hurt so much. The best way is to sit on a chair and lay a towel on the floor in front of you. Put your feet on the towel and try to grab and lift the towel with the toes of one foot. Grab and lift, grab and lift ... don't worry if you can't actually grab the towel each time you try. Just repeat the exercise as if you are holding the towel tight with your toes. Try it 10 times with each foot and repeat whole series 3 times. Your arches will hurt again, but it will be a different pain, more like sore muscles. You are making huge progress.
Do not hesitate to imitate "towel lifting exercise" anytime you remember. When you watch TV or take a shower it is a very good time to stretch your fallen arches and toes.
After 3-5 days of painful running and exercising you are almost there. The pain will get weaker and weaker. One day you will run and realize that you do not feel your arches at all. Congratulations !! Now you can run ... BUT ... don't forget you are still a runner with flat feet. You always will be !! There is no way to get really rid of it, but there are ways to go around it. So think of how you got to this point. The essentials were good shoes, special insoles, exercise and your will to overcome the initial pain while running with flat feet. You should think of it in the "outside of running" life as well. In fact my experience is that the best shoes for walking are worn-out running shoes. This is not always practical so at least wear insoles in your "regular" shoes as much as possible.
If you succeed and run without troubles for several months and suddenly your arch pains get back, maybe it is a sign that your shoes or insoles are being worn-out. I would recommend to intensify your towel lifting exercise, but sooner rather than later I would buy new shoes and insoles. At the end, for a true runner even his or her flat feet are the most precious body parts ;-)