What is the optimal way to climb fastest between two points on a Cessna 172N?
Ron nailed it. (See comment on OP.)
It depends on what you’re trying to achieve.
Climbing “fast” in terms of maximum vertical speed = Vy (best rate)
If the noise sensitive zone is immediately surrounding the airport, you want to use Vy to put altitude under you as quickly as possible.
Climbing “fast” in terms of maximum feet per nautical mile = Vx (best angle)
If the noise sensitive area is a few miles away, you may want to use Vx to get to altitude before overflying the zone.
I gather from the question that the noise sensitive area is right next to the airport.  If that’s the case then Vy is likely your best option.
While accelerating to 100 and then pulling up feels satisfying, just pitching for and holding Vy (or Vx) is more effective.  Peter’s answer explains why this is.
In any case, watch your oil temps (or cylinder head temperatures if you have a CHT gauge) and prioritize keeping the engine cool over noise abatement.  While being a good neighbor is important, taking care of the engine and flying safely is even more important.