Any electronics project that will be put into real-world use needs an appropriate enclosure. For the Ninja Timer, a sprinkler timer cabinet with integrated GFCI outlet is a great choice. It's rugged, relatively inexpensive at around $30, has many predrilled mounting points, included two conuit holes to run wiring, has two interior layers to separate components where needed, and provides a convenient place to plug in power supplies.
You'll also need a power supply cord, such as this to wire the outlet's line side to an external AC power source, following the instructions in the package.
Cut and strip the wire from the AC plug on the 12V power supply. Place it in the bottom of the box, then run the cut AC line out of the box passthrough and then back into the GFCI enclosure.
Screw the wires into the "load" terminals on the GFCI. This allows you to power the supply without need for an extra plug extension.
Screw the GFCI plate back into the box.
There's no need to cut up the 5V 10A power supply, simply place it in the bottom of the box, and route it's wires up through the top layer cutout and plug it into the outlet.
This is what it'll look like with the top layer panel in place.
Plug the 5V micro USB power supply into the outlet as well.
Now you can mount the three boards onto the top panel of the box. Use the 2.5mm nylon screws and standoff and the pre-drilled holes to connect them. Place the level shifter board close to the wire passthrough hole, as these will be pushed outside the box to plug into the digit panels.
For a perfect fit, you may drill out some holes to accomodate additional standoffs and screws.
Once everything is mounted to the panel, screw it into place in the box. Then, connect the three power supplies, as well as the 5-pin and 4-pin JST SM connectors. Be certain you plug 5V 10A into the level shifter board, 12V 5A into the relay board, and USB into the "Native USB" port on the Arduino. (The Arduino will run with power to either port, but we'll leave the "Programming" port open so we can plug the computer USB into in during programming.)
Run the 3-pin JST SM connector cables out of the box, as well as the car horn wire.
You can either run the three Cat 5e / Cat 6 cables to the buttons and button box directly into the enclosure, or use three Ethernet extention cables. Plug them into the J1, J2, and J3 jacks on the Patch Shield.
Make life easier on yourself later by labeling things now! You can trace each run of 3-pin JST SM cables and mark them 0-4, as well as the "start", "stop", and "desk" RJ-45 jacks for the buttons.