First of all, you can send a support ticket and request a USB-serial programmer and we can send you one. The programmer can plug into the 2×3 card edge slot close to the top of the PCB, that way you can program it via USB.
Second, if you have your own USB serial (either CH340, or CP2102 or any other serial), you can also use that but you will have to solder wires onto the serial pins, and the standard procedure for getting ESP8266 into flash mode is to keep GPIO0 pulled down to ground when the controller is powered up. GPIO0 is connected to the middle button (B2), so that needs to be pressed down when you power up the controller and then it will enter bootloading mode.
The USB programmer that we use has built-in auto reset circuit, therefore there is no need to press B2 down when using the programmer.
I am not sure about what you said about GPIO2 — GPIO2 is already pulled up internally on ESP8266, there is no need to add external pullup. Though, adding an external pullup won’t hurt either.