How to route Ethernet PCB ?
I think the routing is at the moment the least of your concerns. Usually, providers of SOMs also provide schematics for a base board. The least painful way to get a working board is to copy those schematics (at least the parts you require). I would guess this information is hidden behind the customer registration item on their webpage.

In any case, the module already contains everything including MAC + PHY (KSZ...) for one ethernet port according to their description, so you probably won't need those. I would guess (according to the pin description) that you would then route the pins to a transformer (1:1 winding ratio) and then to the actual connector. If you want to save on parts, you can get a connector with integrated magnetics (typically called magjack). Again, refer to the module's vendor on how to connect them correctly. The PDF is a little bit on the low information side.

If the track between RJ45 connector and module connector is very short, you can probably (!) get away with simply routing the tracks as you like. For everything longer, you will need to match the impedance closely to 100R Differential or 50R Single-Ended. Make sure that the trace lengths match at all costs.