Don't worry about the insulation, it's normal for it to look tatty and go hard because the heat from the motor takes its toll over the years. Don't try to replace the wires, just use a couple of layers of heat shrink sleeving if there's areas where it's cracking........The more I look at those wires the less I like them - that insulation must be 50,60 .....? years old. It's not old enough to be rubber but it doesn't look like healthy plastic. I may replace it all when (if !) I do the swap - if that's possible without interfering with the windings at all....
As lucien says the name of the game now is to figure out the 3 pairs of wires for the start winding, run winding and centrifugal switch. If you separate out all the wires and measure the resistances I'd expect the start winding to be around 15 ohms, the run winding maybe about 5 ohms and the switch will either be infinite or zero resistance depending if it's open or closed but I'd expect the latter if the motor is stationary unless it's sticking. Please note the values given are a very rough guide and could even be 50% out.