I'm currently building a mow robot that is connected to the internet. The idea is that this way I do not need any buttons on the machine itself. only I big on/off switch which is the green thing on the box at the right side.
As you can see on the picture below there is a netgear attached in the front middle of the box. This allows for connecting the arduino ethernet card to the wifi available in the garden.
As my garden is big (5000 square meters) I have bought a strong wifi transmitter (it stated 4km but I seriously doubt that is possible). I have positioned the wifi transmitter so that I can use in my house and in the garden.
The robot loses its internet connection from time to time but I did not have time yet to investigate the root cause.
I have been considering to replace the netgear by a real wifi router. This would allow me to add one or more IP cam's to the robot. But the netgear is really small and works. If I put in a real wifi router I probably need a bigger box.
If you want to reach your arduino from outside your house you will need to do some network configurations. Which exact steps you need to take depend on the network setup you will go for.
Lost of good luck with your project.