Solder jumper wires to five of the outer pins on each rotary switch, and a normal wire to the central (ground) pin. 

Solder wires to the toggle switch, one each to middle and one side pin.

Solder wires to the pushbutton pins (if being used).

Solder jumper wire to the LED anode pin, and a 330 ohm resistor to the cathode pin; then a wire to the resistor.

Solder jumper wire to BNC signal pin (central pin), and a normal wire to the BNC ground (side pin).

Jumper wires make it easy to connect to the Arduino I/O pins, but there aren't sufficient ground pins, so you need to use normal wires, then solder those together.

Any exposed regions can be shielded with heat-shrink tubing if there is any concern of movement/contacting other bits.

Plug the collated ground wires into the Arduino Ground pin, and plug the other jumper wires into the correct I/O pins:

Toggle switch - 0

Pilot LED - 1

TTL output - 2

Rotary switch 1 (ontimes) - 3, 4, 5, 6, 7

Rotary switch 2 (frequency) - 8, 9, 10, 11, 12

Download the .ino file and upload the code to your Arduino Uno.

Drill holes in the case for each component, keeping in mind that there is not much room in the case, so place the hole appropriately to allow room for soldered wires etc. around the Arduino pins. Push the components through and bolt in place using provided nuts (the LED will need to be glued in place; I used superglue). 

Depending on which case you have purchased, it might need to be glued shut. If so, be sure to test the TTL driver works as needed before sealing shut. You can still access the Arduino Programming port after the case is sealed, so the program can be modified after assembly; you just want to make sure everything is properly connected.

Push the potentiometer knobs onto the rotary switches and apply markers (I used cut up labels) to identify switch positions - this is easily done by connecting the TTL output to an oscilloscope or LED and noting which positions produce each set firing pattern.