Before you get too deep into this project you will need to swap the alternator from the old engine into the new engine as the new engine has a low output that will not be able to keep up with the mower when it is running plus keep the battery charged
While the blower housing is off the old engine note which wires went to what items
One wire goes to the coils , one wire goes to the carburettor and one wire goes to alternator ( via the rectifier )
Then check the old plug and put tags on the MOWER side so you know which wires to join to what
To do the job properly you should remove the terminals and swap their position in the plugs.
leave the magneto wire disconnected at the magneto and when you have finished with the plug check that the magneto wire is either open ( not power or ground ) with the key in the on & start positions and ground with the engine in the off position.
A good mower shop would have pulled the plugs apart & swapped terminals around so it would be plug & play in future or fitted a cross over sub loom so the mower & engine plugs remain the same .