Neal is right .... old school used green board drywall. It is NOT waterproof and barely water resistant. The "minimum" approach is concrete board with waterproof strips thinsetted into the joints for best protection. Waste the weekend because it is NOT worth the future mold problem.
As to your bullnose, I would use grout on the edges to fill the gap. Lay your bullnose just past the edge then grout. Make sure ALL your grout lines are sealed three times before using the shower. I use a cheap kids art brush and work the sealer into all the grout lines.
Good luck .... and lose the drywall!