Today I added the USB connector, as described on this Imgur gallery. I'm going to find a more elegant solution to some issues, but it does the job for now.
I used a female USB plug and soldered it to the PCB points using a four wires. They are currently unshielded, but that's not a problem for a low-speed device like a keyboard. Maybe I will find a more elegant solution in the future. The USB plug was then hotglued to the back of the case and will hopefully stay there for a while.
Once connected I could access the MAME menu by pressing the [TAB] key. The arcade buttons was accessible from there, and configurations automatically saved.
Below is the finished solution. That'll do for now. I removed the UART wires as they will be replaced by a BTLE wireless UART module later. The USB keyboard was left inside the arcade in case I need to reconfigure some keys on a later occasion.
Super Mario Bros works perfectly, and I'm looking forward to spending hours of fun with this arcade!