Last time I went to Astore in 2018 the road from KKH to Astore was under construction. If the road construction activity was completed than condition of road should be fine, otherwise you may encounter some rough patches.
That road however, finished or under construction, is amongst the scariest roads I have traveled on. The narrow road with loud sound of violently fast river can make driving a tense experience. Road to Rama meadows is also OK, but not great. You can get there on your own car without too many issues.
I would not advise you to go up to Deosai on your own car, if you care about your car. I have seen people in Deosai in their Suzuki Cultus (old model) and other small cars, but I wouldn't recommend it. Honda City can get there, but there are patches and some small streams (which depend a lot on weather each year) where cars like Honda City will hit the ground. Better to get 4x4 from Astore in my opinion. Or you can even get 4x4 from Gilgit and leave tension of driving altogether.
