If you don't plan on spending some time exploring, climbing or doing some photography on or around Looking Glass Rock, then there's not much reason to take the dirt road out to this spot. The road is dirt and well maintained until you take a cutoff to your left right as you're getting to the western boundary of the rock form. There are primitive campgrounds scattered all over the area surrounding the butte. There are some sites that do require a high clearance/4WD vehicle in order to get to as you'll be driving over uneven slick rock and some moderately deep sand. There are no amenities so bring everything you need and then pack it out!
As for adventures we did while in the area, we visited the always popular Wall Street climbing area to the west of Moab on Potash Rd. If that's too busy for ya you can also go a little further up the road and explore around Day Canyon where we also climbed the tower in the pictures, Boognish Tower. And of course if you're looking to just spend an afternoon or a night around the arch then the East Rib Route on Looking Glass Rock is an amazingly easy beginner climb (albeit slightly run out) that has a spectacular ~120' completely free hanging rappel through the window at the top of the route. Certainly the best arch rappel I've done in the area and definitely one of the more unique beginner climbs in the Moab area.