The different times of day do look pretty different, with the 3 main sections being morning, day, and night. They are dependent on real time, but if you don't want to wait you can change your system clock to whatever you need it to be. Teddiursa's supposed to be available during the day, which I think is anywhere between 12 and 8 pm. As for the weather, it'll be pretty obvious if it's rainy or windy, but if its sunny then the screen will be significantly yellower than normal once morning is over, and clear weather which you'll be looking for is just the normal color scheme. If you don't get the right weather and you don't want to wait for that either, you could try downloading the weather mod to be able to change it in game, or just change your system date since it generally changes weather every day.