Will this antenna even help?
The link that you provided does show that it can work in the 2.4GHz frequency. And for the full duplex doubt you have. There can be many ways you can fix that. A time tokening is a most simple method of doing so. Many wifi modules have dedicated antennas for transmission and reception. Even if the antenna is one, the wifi module that you speak of is a simple one that can be interfaced to a MCU(so probably UART or SPI is the interface between the WIFI ASIC and MCU, so no fancy high speed).

The mnodule itself has a patch antenna on the PCB. So, how do you intend to connect the u.FL to this module ?