How do I connect Arduino to RS232 via USB?
This is effectively not workable.  USB is not a peer-to-peer standard, but a host/device one.  Both your Arduino and the USB-serial converter are intended to be devices, so they cannot communicate.  While it would be possible to use another microcontroller acting as a host to bridge to serial that would be pointless, as the actually central problem of RS232 vs. logic level serial would still need to be solved.

Instead, what you want is an inverting serial level shifter - essentially a board with an IC like a MAX232 or one its many successors.  You would either connect this to arbitrary digital pins of the Arduino and use a Software Serial instance, or decide that it is okay to block usage of the USB port by a computer while this is being operated, in which case you could connect it to the D0 &amp; D1 pins used by the on-board USB interface (there are typically resistors in between the chips on the board to support this overriding).

Ironically, in a way what you want is one of the original Arduino boards that had a serial level shifter and 9-pin connector in place of the USB chip and USB connector of modern boards.  But then, you probably want something with an ATMega328p rather than the earlier, more primitive ATmega chips used on the original boards.  If you can still find a board of the original design, upgrading the chip may be an option, or it may be that your task is simple enough not to require it.

There have been many other non-Arduino embedded controller boards released with native RS232 ports as well, especially those dating from around a decade ago might give a good combination of fairly modern capability with not yet having abandoned RS232 in favor of USB.