Transportation in Tokyo is all about the Metro and JR trains. Purchase and load Suica cards at your first station to make getting around much easier. And bring your child's passport, since it's required to qualify for a reduce-priced Suica card.
There's lots of walking involved during this day trip. Even if you take public transportation, the gardens and temples are large and the only way to see them is by walking. It's normal to walk 20,000 steps a day in Tokyo.
This is a great trip any day, just note that weekends in Harajuku are far more crowded than weekdays.
Spring, summer and fall are probably more enjoyable times to enjoy Tokyo than the colder, winter months.
A hotel in Tokyo can be as expensive or inexpensive as you want it to be. Other than the hotel, the only expenses associated with this itinerary are public transportation, meals and games/toys. None of these attractions have an entrance fee.