It is possible to walk a limited way up the (very steep) stairs of the main prang, which gives a reasonable view of the Chao Phraya river.
Despite its name (coming from Aruna, the Indian god of the dawn), the best views and photos of Wat Arun are in the evening with the sun setting behind it. There are some restaurants on the opposite side of the river that are good for watching this, though you'd be lucky to see the image of Wat Arun that's on all the postcards - that of the red sky sunset with the sun setting directly behind the temple. Sunset is around 6pm - 7pm all year round in Bangkok.