A lot of good suggestions have already been made, let me add a few:
- in your settings, potatoes might not be the best choice of growies, since they require quite a big container to get a one-dinner-harvest. There are better choices for growing on windowsill, starting from culinary herbs, through chillies, micro-dwarf tomatoes, chives, etc.
- if you insist on growing potatoes, wise thing would be to stick to super early varieties that grow fast and are smaller in size.
- thinking out of the box, I would examine possibilities to grow potatoes actually outside the appartment - on any roof available in vicinity, someone's lawn, or I would join a community garden in my city perhaps. All you need is to ask someone "may I place this bucket here?" - and there you go, you have an instant growing place ;)
- there is a separate category of potato growing techniques - growing them from seeds. So called True Potato Seeds (TPS) can be sown in a very small pot, and at the end of the season you will get a few very small potatoes :) Normally, you would use these as seed potatoes in the following year, but they are perfectly edible of course.
A picture shows early variety Charlotte, grown in 5 gallon bucket, in the city, on the terrace.