You don't mention how young the kids are but I assume pretty little. If that is the case, I agree that the limo-bus is your best bet. You will be traveling during rush hour so train stations will be chaotic. Let the bus take you straight to the hotel. Luggage goes in the hold under the coach and is loaded/unloaded by the driver. Well worth the cost. You buy tickets at a counter in the arrivals hall.

You don't say where you are coming from so it is difficult to guess how bad the jet lag will be. Assuming someplace in the US, it will be substantial but manageable. Try to encourage a nap during the flight but don't sleep the whole way so you arrive tired and ready to sleep. Then, after arrival, try to keep everyone up and active until a normal bedtime. They will want to collapse in bed immediately upon arrival. Go out for a meal or a walk around the neighborhood. Look at the lights, take some pictures, whatever, just don't go to bed until 9:00 pm or later.

You, and the kids, will undoubtedly wake up very early the first morning, perhaps before first light. Many use that early rising to go to the Tsukiji fish market for the early morning auctions but I think that is silly, especially with little ones. That first day, get out and into the sunlight as much as you can and again avoid the urge to nap. Go to bed at a normal time again and you should get adjusted to the local time pretty quickly. If the kids normally need an afternoon nap, try to limit it to an hour or so. They need to adjust to local time quickly.