I do minor refurb on those Rain King sprinklers, I absolutely love them. They do a fantastic job of watering the lawn. I have two in active use. One of them being a childhood sprinkler, but albeit it's not a very happy camper at the moment.
There's two different versions of these sprinkers, one with a large nut holding the bottom on, these are very early, late 40's. The next iteration had three screws holding the bottom on.
The biggest killer on these is freezing. It doesn't seem to matter if the sprinkler is completely drained, at the end of the season, drain it, and put it someplace where the tempature is relatively constant. Good example would be the top of a full size fridge in the garage or a shelf about halfway up the wall.
If it's leaking at the bottom, the gasket can be replaced, then resealed with silicon. Carefully remove the screws holding the bottom on,cut a piece of gasket material to fit, silicone and reinstall.
If it's leaking at the top, that's more difficult. Remove the screw holding the red top on, under that, you'll find two more screws, remove these and gently pry the silver top off to gain access ot the head. Be exteremely careful of that gasket that's directly underneath the head, this is not an easy replacement, and you might have reuse it. Once inside the head, take note of where the teeth are, and used to a sharpie to mark the location of the teeth (VERY IMPORTANT). Gently remove the gasket, taking care not to break it. Then take extremely fine sandpaper, and clean where the gasket was sitting, replace the gasket and run a bead of a very thin bead of silicone around the gasket. Replace the silver top, tigthen down, and wait on the silicone to dry.
*OPTIONAL
While the head is disassembled, you can remove the arms and clean/polish them. There's two different models of these sprinklers, one with the screw in jets, and one that dont. There's also two arm gaskets that could be replaced, but if they aren't leaking, leave them alone.
To remove the arms, if your model has the screw in jets, (like the one pictured in this thread) gently removed them, and push up on the bottom of the arm, it should pop up. You can use Brasso or other metal polish to clean them up.
Final note-
If the sprinkler pops up and stalls, it's toast. The main bearing is completley shot.