Backpacking is allowed, just not encouraged, in Arches.
The thru-hike of Courthouse was from the highway up near the turnoff to Island in the Sky down to the highway near town is very nice. If you have a bike a self-shuttle is easy.
Exploring the Courthouse Towers area is fun. Near the southern end of the Park Avenue trail there are several scrambles up to the top, and at least one way back down on the north side towards Sheep rock.
Another great scrambling area is Herdina Park, off the old entrance road. You can hike north from the road and find a way to Eye of the Whale Arch through the middle of the labyrinth, but it might take a bit of wandering and problem solving. It's like a little Fiery Furnace without the permit nonesense.
Hiking Salt Wash from Wolfe Ranch down to the Colorado is fun, but you need a car shuttle or long bike shuttle, and you have to swim the river. Carried a wetsuit along to do that hike in February back in the day.
I've always wanted to explore Lost Spring Canyon, but never got around to it. That would be a fun overnight.