My rear screened porch has some has 2x2s that hold the screens in place and many of them are rotted from moisture.
It looks like they are plain old 2x4s that were ripped down (though I'm not sure) and then painted (poorly) with exterior paint.
What type of wood and should I use to replace them? I do wish to paint them to match the rest of the house. Or can I use any good quality wood and prime and paint with a good paint?
As long as you make sure that the wood is isolated from the source of moisture as much as possible (or better still remove the source) then any decent quality wood properly primed and painted should be OK.
I would also consider getting the wood treated first.
Cedar would be a good choice, if you want something untreated.
Most timber that has been painted is done after the fact and typically the areas that are likely to soak up water are not accessible to the painter, such as the underside of the timber. This is typically also where water can settle and therefore will get soaked up by the timber which creates the rot due to not being visible and therefore not exposed to sunlight to dry out. When installing new timber in exterior locations it is best to paint it after being cut to size to ensure that in unseen places a coat of paint creates a water barrier. Most hardwoods are best for exterior locations, but nearly all timber can be used if properly painted as @ChrisF says.
