The homestay usually lasts from 4-8 weeks. The new Nagoya JET will be placed with a family somewhere near Nagoya. This is a time to get used to being in Japan if you haven't been here before, get over JETlag, and find your new apartment and get settled. A lot of it depends on how you and your family interact and how well you both try to communicate. One thing you will quickly realize from spending time with your host family and learning about other JET's host families, is that despite all of the stereotypes, Japan and Japanese families are not homogenous.
Every family is different, just like every JET is different.
Try to make the most of the situation. It is a good chance to find out about Japanese people and culture and a good chance to make friends who might also be able to help you a lot later. Eventually, you will need to find an apartment in Nagoya. To be honest, this is where many new JETs get very frustrated and feel like they have little support from Nagoya City and your new supervisors. If you get here quick and are lucky, you may also have a chance to look at and grab the apartments of departing JETs and even get some good deals on furniture, kitchen utensils, etc.