I think the chances are pretty high. Others may have had different experiences but you're going to be taking it at rush hour. A lot of business types do fly in on early morning flights and take N'EX in to do a day's work... that's in addition to all the regular vacationers and tourists.

You could still get a seat yourself, but there may not be space for your luggage and you may have trouble getting through a crowd of people with your bags at Tokyo (assuming that's where you're going). I've had that happen. Not just the seats but the luggage racks can be totally full. I have not gone during Golden Week or anything, this was just at rush hour.

I have taken the limo bus and much as I love trains, I much prefer the bus to Tokyo. Or the Skyliner, if you happen to be going somewhere that's easy to get to from where it stops. But the bus stops at T-CAT which has not been crowded the times I've been there, even at rush hour.

Of course, if you really need to go to Tokyo Station or Shinjuku, then N'EX is still probably the best option. Maybe think about a green car if you can afford it - they rarely fill up on any train.