Wiring – How should I connect this old fluorescent lamp

Most likely the second white and second black wires are the connection to the switch itself. If so, then what you've got is power line from circuit box to light, switch line from light to switch. (The alternate method is power line from circuit box to switch, switch line from switch to light - which would have been less confusing for you but he probably was dealing with old work by the sound of it.

How it works in your configuration (if I'm right) - imagine the electrical current like a flow of water. The power comes in from the breaker box on the black line (aka the hot line). The power then runs to the white line and down to the wall switch. When the switch is on it then runs back up the black line to the light. You connect black to black, and the power flows into the light, then out the white line. You connect white to white, and the power flows on down back to the breaker box, and out.

So - don't touch the second white and black. Just wire the light's white to the lonely white and black to black the same way.

My guess would be that because the fixture has a metal frame, the ballast is grounded via its contact with frame. If the frame were plastic, you would need to use the terminal on the ballast.

If you're converting it to hardwired, you should definitely connect the ground wire to the grounding screw.

You're also going to need to cut a hole if there's no punch-out available and make sure to use a cable clamp/strain relief suitable for using with romex.