Neutral and Live both hot when switch is turned off. How do I connect my new light?
The way you describe it, the receptacle has:

one hot and one neutral (single wire) when switch is on
two hot (single and triple) when switch is off

That would be functional (as in the light would turn on and off correctly), but dangerous (all the time) by modern standards.
The light would go between the three black wires and the single black wire. (two hots: no current, light's off) (one hot, one neutral: current, light's on)
It is possible there simply was a miswire at some point between neutral and hot.
Given the current state, if the ceiling light is high enough to not be a shocking hazard and you don't feel like fixing it all, all I can offer is:
Turn off the main breaker at the panel? That would turn off electricity to the whole house/unit. Use your cell phone flashlight to connect the ceiling light (or sunlight).
The only alternative is to identify where the miswire is and bring it all up to code, but that would be more effort.