How to move from arduino to physical prototype
The best thing to do is, of course, to use a PCB. But it doesn't need to be a custom PCB; there are plenty of pre-made PCBs that are usable for approximately 98% of designs. Here are just a few:


Adafruit Perma-Proto boards are connected just like breadboards, so direct transfer is possible.
BusBoard proto boards come in a ridiculous variety of configurations, including enclosure kits, and they even have SMD boards if you're up to it.


For switches just use a panel mount form. They come in everything from simple to robust to ridiculous.

LEDs can also be panel mount, or you can use a light pipe to make a LED on the PCB visible from the outside.

For power you can use a USB connection or a separate power jack with a voltage regulator. If you also need a serial connection then you should consider a pre-made serial module, and either use its built-in USB jack or a separate panel mount USB jack.

The servo can be mounted inside the case if appropriate, or can be connected to a PCB mount or panel mount connector instead.

Naturally the MCU itself should also be placed in a socket directly on the PCB. Don't forget the decoupling, the oscillator, and the ISP connector.

As for the circuit itself, you can get rid of the pullup resistors on the switches if you use internal pullups instead. And 100ohm is way too low a value for switch pullups regardless.