There are several places in Arches NP that would put you with an easterly view toward the rising moon. I would recommend that you drive through before the fact to scout out your vantage point. It has been a number of years since I visited there.
Additionally, If you want a long time exposure, or multiple exposure opportunity take hwy 191 south from Moab toward Blanding. Appr 30 miles, as I recall, you will come to a turn off to the right. (Just past the LaSal jct. about 8-10 miles.) Take the right turn off and proceded wnw... when you come to a fork in the road take the left juncture. This will take you to the needles overlook. At the needles overlook you will have a southerly and southwesterly view. A daylight shot of the needles followed by a second exposure with the moon above, however many hours later that would be, would probably be an unusual and interesting shot. It would certainly be different then all of the arch shots that usually are the product of that area. If you are up to camping, there were places that were available when I last visited there...a truly memorable night in my experience.
Good luck and I hope that you share your experiences when you return.