From what I can tell, you need to hook the mains up, and a lamp as shown in the drawing you made. The rest can possibly be left unhooked, but I don't know for sure.
The square box is a ballast, not a battery. You need a ballast to provide inital high voltage to start the fluoresent tube and then to limit the current though it.
A small fluoresent fixture isn't that expensive new here in the US. If this was found in a shed, it may or may not work. Ballast degrade with time.
Using this with all the extra wires is going to be more complicated then just buying a new fixture with wireing instructions. That goes double for someone with limited experiance in electronics and electricity.
Also, even with the battery, if its wired correctly it won't come on when the switch is turned off. There should be an extra wire to control the battery backup. You can't have the batterys draining when somone simply turns the switch off.