How to open this fluorescent light fixture

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.

Besides the bulb, there are two other common failure points: the inductive ballast (which limits current and, for long tubes, is a step up auto-transformer) and a preheat/starter switch (not a capacitor).

Both those components are replaceable, though it may be difficult to find them. In the 1940s through the 1970s, they were available at every hardware and electrical supply store, but since standard florescent fixtures have now been completely obsolesced by compact florescent (CFL) and LEDs, compatible components are now uncommon.

If you can't find parts or they are expensive, this might be a good opportunity to replace the light fixture with a modern one. It will be about three times more efficient, quieter, more reliable, and more easily repairable. They should be quite reasonably priced too due to significant competition.