Hachiouji-shi,
You're starting to get me a bit worried. The B737 is not exactly the fastest thing on wings, but Mcrit would be well above M0.615, that would 'Max out' the 'useful' cruise at about M0.65. I recall that it's about M0.72 to M0.73, but it's 23 years since I last flew a B737-300.
Why would flying above Mcrit be a bit uncomfy for the passengers? They wouldn't even notice it! I've operated about 16,000 flights on jets above Mcrit and I haven't noticed it yet, maybe I'm a slow learner.
Actually, I have to take a step back, and say that they MIGHT notice it - visually. If you have a wing seat, the sun is low in the sky, and you're looking along the wing in the general direction of the sun, the shock waves can be highly visible, and very pretty to watch. They don't let us fly fast enough these days to see the big beauties that we saw on the B707, but, in the right seat with the sun at the right angle, you might still see the occasional itty-bitty fuel economical shock waves of our times.
Regards,
Old Smokey