Yes, you have plasterboard in lieu of lath. I've found that often the corners and ceiling to wall are backed with wire mesh and don't crack as much as the flat seams between the sheets, which is what you're seeing.
It would be faster and you'd likely get a better job if you overlaid it, but if you're doing it yourself it's probably easier to tape the cracks and skim coat it.
First I would clean it well with something like tsp.
Then I would clean the cracks with a utility knife or a stiff putty knife to get the loose stuff out. Or maybe a stiff, narrow brush? You just don't want chunks falling out while you're trying to tape it.
I would tape it with that sticky mesh tape instead of paper tape, it tends to hold a little better, but it is slower to finish.
I would buy general purpose mud (aka joint compound). The powder mix is ok and has it's uses (like thickening primer), but it's definitely slower and harder to mix, and if you're doing an entire ceiling of cracks you'll use quite a bit.
Joint compound is sold in plastic bags in cardboard boxes, but you can squeeze it out of the bag into a clean 5 or 2.5 gallon bucket with a lid and it will last for months. If you are using a lot you can use it straight from the bag into your tray, but if it dries some in the bags little chunks will fall in the clean mud and be a PITA.
I would start with a fairly flexible 6" drywall knife and cover the tape well without leaving a lot of extra mud on the ceiling.
Knock down the ridges with a stiffer knife. Some of the mesh tape will be exposed so you don't want to sand it yet.
Switch to a topping mud and put on subsequent wider courses with larger knives. 8"-12".
Sand as needed, hopefully minimally.
It's going to be hard to get it really smooth like a nice old plaster ceiling, so you will probably want to add a little texture, maybe just a stipple from thickened primer and a big nap roller, but heavier textures are used to hide problems (like your ceiling). When I'm doing drywall work on my own house I usually reach a point where I decide that perfection is not worth it and it's "good enough", and then I roll on a little texture.
I bet the roof and the subsequent attic insulation added quite a few cracks to the mix, so it's good that work is complete.
Also, take the time to roll out and tape down some thick floor paper, you won't regret it. And yes, most of the work can be done from a two step ladder or, even better, two ladders and a plank. You can also get a 30" wide x 6' long adjustable height rolling scaffold for indoor work; they're narrow enough to roll through a doorway (we call them Bakers around here). You can find used ones for a hundred bucks or so.
