In fact your adapter can not and does not use 8139too this is for ancient RTL8139 which was common when motherboards did not have ethenet and these were plugged in the PCI slots. Since yours is an internal adapter it was attached (i.e., hardwired over motherboard) to the PCI bus internally so you can not change the slot as the article defines. Besides this is for obsolete devices and when Plug-and-Play was Plug-and-Pray :). I don't think none of these will work for you.
Besides though lspci lists your adapter:
04:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168B PCI Express Gigabit Ethernet controller (rev 01)
There's no corresponding dirver for it present on your system. You need to download the dirver, the corresponding kernel headers file and recompile the driver which is here:
http://www.realtek.com.tw/downloads/downloadsView.aspx?Langid=1&PNid=13&PFid=5&Level=5&Conn=4&DownTypeID=3&GetDown=falseCheers,
K.