Hi There,
Arch pain isn't uncommon but if not addressed can lead to more difficult issues to treat such as chronicplantar fasciitis or heel spurs. As a physio, a keen runner and a hypermobile overpronator i have had similar issues personally, and treated many clients with the same. If you are going to wear corrective devices or sports shoes that change the position of your foot when you are exercising it's a good idea to get a good sports podiatrist to examine your foot position with and without the devices to make sure they're not going to do more harm than good. Then it's a matter of getting used to them... I often explain to clients that getting used to orthotics can feel like you're walking on tennis balls for a few days, usually wear them in for half an hour each day and increase by 15 minutes daily until you can comfortably wear them all day.
For you, this sounds like a reasonable option in that you're only trying to wear sports shoes for a short time and doing impact work in them straight away. You would be better off getting used to shoes/orthotics that support your arches all the time.
As far as exercises go, there are a few things that can help strengthen and stretch the arch of your foot. To stretch it out I tend to recommend rolling a golf ball around under your foot to work out all the "crunchy" bits along the arch. For strengthening the exercise is generally referred to as "doming" (it's a bit tricky)... you sit with your foot flat on the floor and relaxed, then you try to JUST lift up the arch of your foot gently without crunching your toes into the floor. the muscle which does this is called tibialis posterior and you can feel it working if you rest your fingertips just on the inside of the border of your shin bone about 1/3 the way up your leg (google the anatomy.. it's easier if you have a picture!). you can also do other strengthening of the intrinsic foot muscles by practicing scrunching up a towel with your toes, or picking up marbles with your toes. Also stretching the calf muscles is generally necessary.
That's quite a long winded response... hope it helps.
msk101