The idea is that i can just plug this small device to an ethernet cable, to see if there is a connection to a switch on the other end or such.
You can use it to see if there's a connection to the switch but you won't see the speed as this device is not able to handle Gigabit Ethernet.
Not to see if there is internet or anything else.
That's a much more complex question anyway. There is no general way to see if there is Internet. For example it might be possible that I get an Internet connection if I connect my notebook but don't get one if I connect that Arduino device (might be filtered based on MAC address or anything similar).