In summary, a force of 1 Newton x 3 meters is equivalent to 3 joules of energy. This energy is used to change the velocity of a 1kg mass, but it will not result in a velocity of 1 m/s. It is possible to calculate the speed of the mass using the SUVAT equations, which relate distance, time, and acceleration. However, the units of "m/s/s" do not make sense for describing uniform acceleration.
You did 3 Joules of work on the mass. Energy is conserved so you know what its kinetic energy is. You can calculate its speed from that... and it won’t be 1 m/sec.
Or you can calculate the acceleration (you know the force and the mass) and use the equation that relates distance, time, and acceleration to calculate the time it took for the mass to move 3 meters. Once you have the time you can calculate its speed... and it should agree with the other speed calculation.
This would be a really good time to Google for “SUVAT equations”