Dry wall the ceiling first---that is required by most building codes. It is a fire break thing. Codes usually specify 5/8" sheetrock for ceilings and 1/2" for walls.

Them do the walls and tape/mud all joints. Since the ceiling/wall joints will be hidden, two coats of mud/sanding will be sufficient.

Then install the bead board.

The required fire blocking is provided by the top plate of the stud wall and there is no particular reason to use 5/8" drywall on the ceiling unless you need additional sound separation or greater resistance to sagging in a high humidity space.

Drywall is usually hung on the ceiling first so the last foot or so can be left unattached and the wall board can then be pushed up tight against it (with a foot lever tool) so the ceiling board is trapped.