For people who have been to Tokyo and/or speak Japanese, the price seems silly. As a cheapskate myself, the price is tough to stomach. But having traveled with 2 kids (in economy), there is a serious value proposition in having someone meet you and take care of getting you where you are going when you are jet-lagged. [Note: I've never stayed at the Shangri-La.]

When my kids were younger, we took an MK Taxi a few times, and it was worth it. Lately, the NEX works fine for us (we also arrive a bit earlier ~6pm). But my family speaks Japanese....

You'll have to balance your ability to navigate Japan for the first time (it isn't hard, but it will take some amount of effort) with jet lag and family.

Not knowing where you are coming from or how well your 9 year old travels makes it harder to answer. First time to Tokyo and later in the evening makes me say hit the easy button and pay the price to take the car service.