I'll simplify it to be a gun instead of a rocket, and it's fixed to the front of the pipe, pointing backwards.

You fire the gun, and it gives momentum -p1 to the bullet. At the same time it gives momentum p1 to the gun and pipe. Exactly the same as recoil in a normal gun.

Clearly the pipe's now moving because it has momentum p1.

Eventually the back end of the pipe and the bullet collide, and the bullet becomes embedded in the back of the pipe. They were travelling in opposite directions, with zero total momentum (-p1 + p1 = 0). Now they're a single solid object with the same total momentum - 0.

The pipe has 0 momentum and has therefore stopped moving.

the bullet is not itself pressurized , its molecules will not break apart and expand throughout the pipe.

the air moves throughout the pipe , some of the air moves into every cubic inch of area inside the pipe.

because the air pressure itself is equalizing.

because the tank is at one end of the pipe most of the
air in the tank will move toward the other end of the pipe.

and since the pipe is 60 inch diameter and the tank is 58 inch diameter there will basically be just air equalizing at the end where the tank is.

all the other air moves away from the tank.

and it is this movement that moves the pipe.

in my last post I calculated the momentum to be
99,557 kg-m/s

and this movement of mass inside the pipe causes the
pipe to move.

so if you move this mass fast or slow doesnt matter
as the total momentum is the result of P = m*v

in your bullet anology you would still have the reduction
in velocity due to air resistance as the bullet travels the 500 ft distance to the other end.

then F = m* (less velocity)than you started with.

the pipe would still be moving.

3/4 inch of dust build up on the moon in 4.527 billion years,LOL and QM is fantasy science.