How do I connect this SPI serial display to an Arduino?
That kind of connector is most commonly directly soldered on to a PCB.  You can buy adaptor boards with the various different pitch pads on them for soldering it to.



Also there is no mention of what chip the screen uses (I am unfamiliar with "XS057" as a chip, so that is probably the model of the screen not the chip).  At that resolution it's probably an ILI9163, but without being sure all you can do is stab wildly in the dark and try different drivers.