Putting air into the transmission would only cause oil to possibly be forced into the suction side of the pump, so yes helping the system out a little.
I wouldn't recommend doing it again as it very easy to blow out a bunch of seals and gaskets doing that.
Rebuilding the pump is not an option, you just replace it.
Need to know what model to give you the right pump P/N.
But you need to check the system out a little more before just throwing a pump into it, as it could be an issue with the control valve (rare), relief valve (not as rare), damage to either the three point piston, cylinder or rings (very common), or yes the pump.
If you follow the low pressure line from the pump and you run into a rectangle block, you can pull the plug out of the side of the block and install a pressure gauge and test the pressure, pressure will be almost nothing until you engage the three point control.