There are a couple of ways you can solve this one...
When it came to painting this monster here, there was no way I was going to do it with a brush.
Some folks have suggested being careful how you join the corners. That is a good tip, but it creates tons of additional work. You can use a small strip of paper over the joint to cover it (the exposed inside foam) even quicker.
Most spray paint will melt the foam. There is the H2O stuff out there and it works quite well for NOT melting. I've used it before, it's just expensive. I've put that down over the exposed edges of the foam board and then spray painted the rest with regular spray paint and had no problems.
You can go the white (PVA) glue route too. Thin it ever so slightly with water and coat the exposed edges of your foamboard. This takes time to dry and you might miss some places in addition to possibly creating unwanted texture along the joints.
And for those interested, here's the whole process on how I built that cathedral:
http://fromthewarp.blogspot.com/search/label/cathedral