If you are driving directly from the North Rim to Arches, just get over to hwy 191 and go north toward Moab, where Arches is located. However, if you have time, you are not far away from Zion and Bryce, as well as Capital Reef national parks.
But if you are visiting arches, you almost need to visit Deadhorse Point state park, and Canyonlands Island in the Sky District.
Decent 2-lane roads all the way to Arches and beyond. If you head out of arches up to I70 and then west to Hwy 6, and then north to Price UT, you can head up 191 out of price and continue on up to Jackson, then Yellowstone. (NOTE: A serious climb out of Price UT toward Vernal.)
My recommendation for heading from Yellowstone to Cody is a long way and that is exit the northeast exit of Yellowstone and head out 212, aka, the Beartooth Hwy. At the end you are in Red Lodge and you can head south from there to Cody.
Plenty of camping around Arches in the BLM campgrounds; however NO hookups. Also, plenty of areas to disperse camp in the arches area.
Also, some nice campgrounds outside of Yellowstone on the Beartooth; however, some take no reservations. At the end of the beartooh, just before Red Lodge is a nice campground at Greenough lake. In fact, there are about four campgrounds in that immediate area; all part of the Custer NF.
If you exit out of the east exit of Yellowstone you are in the Shoshone NF and there are some campgrounds along the way to Cody.