This is done via the clock in Nancy's phone. Click on Menu, Clock, then change the time to the time you want to go to and click Alarm. Nancy will only do this in places where it's acceptable to sleep, so that means her room at the ryokan (mostly).

The action of the game only takes place in the evening when Nancy is at the ryokan. During the day, she works at a local school teaching English. This means that if you try to set the alarm for 7 a.m., she won't do it; she says something like, "No, I'll be at school then." So she'll only let you set the alarm for times between 7 p.m. and 4 a.m. Be aware, though, that you can't use the trains or call people after 1 a.m.

As to when you change the days, that's up to you. Change the time when you've done everything you can think of to do, or when you want to get up after everyone else is asleep, or when you want to have access to other people again. The change of the days only matters for the first three days, when Takae will have new lessons for you. After that, you can change whenever you want.

In general, time passes a little oddly in this game. It doesn't move at all when you're playing pachinko, working on a puzzle like a nonogram or one of Takae's lessons, or talking to people. However, it jumps forward rather quickly when you're walking around, so that it looks like it took you half an hour to go from your room to Takae's room. This shouldn't affect you too much, but if you want it to be a specific time, the best way to get there is use the alarm as noted above.