Last time I rode a coaster brake was maybe 45 years ago? Anyway, as I recall, the usual method was to roll the bike backward a bit to lift the appropriate pedal to the desired starting position. That was with flat pedals. Clipped in with coaster brakes would be kinda weird, IMO.
Another method, advocated by my father, was to stand beside the bike, put the appropriate foot on the pedal on that side, push off with the opposite foot to get rolling, stand on the foot on the pedal, then swing the free leg over the bike. You can do the reverse in dismounting, as is commonly done in CX nowadays. I still mount and dismount like this on occasion on my road and mt bikes, just for the heck of it.