You can cut off the connector and wire directly to the cable. Leave enough wire on the connector so you can determine which wire goes to which contact on the connector. They may not have a universal color code like USB.
Each analog multiplexer connects one of 16 pins to an Arduino analog input. In the diagram each of those 16 pins has a pull-up resistor. Those are only necessary if your "sensor" is a variable resistor: LDR for light, Thermistor for temperature, PDR for pressure... In those cases the other end of the sensor goes to Ground and the sensor/resistor combination acts as a voltage divider.
If your 'sensor' is a potentiometer you should leave out the pull-up resistor. Connect the ends of the pot. to +5 and Ground. Connect the wiper of the pot. to the input pin.
If your sensor is some other form of voltage source (0 to 5 volts) you should also leave out the pull-up resistor.
Please note that the "volatile byte" arrays are separate from the "static byte" arrays even though they have the same name. The static ones exist only to the code inside check_switches(). All the other code uses the volatile ones. You should probably get rid of the static ones.