Well, I fixed mine back in the 60's with nothing but a screwdriver and an adjustable wrench so no, you don't need the Park tool. However, I don't see a lock nut, and I'm not sure there is a proper lockwasher there. Hopefully you have both or can get them (can always just buy a complete set if the parts are unevenly worn or pitted. See the pic below. Once bearings are greased and the BB is assembled you can just turn down the adjusting cone by pushing around the notches until you feel resistance. Turn it back about 1/8 turn and tighten the locknut. Should have no play but turn without binding (don't expect extremely smooth with this level of quality). If too loose or too tight just loosen the locknut, move the cone very slightly in the needed direction and retighten.

The reason you don't need a tool is that the lockwasher is supposed to have a tab in it that fits into a notch in the spindle portion of the crank. That prevents the cone from turning when you tighten the locknut. As for the cups, nothing fancy needed - even in the shop we generally just used a block of wood and a dead blow hammer.