You can do this with the old solenoid that you took off, fix a wire onto the small connection to the solenoid, now connect the body of the solenoid (steel part) to a pole of the battery, it makes no difference's which one for now because you are just testing it, and the the small wire from the solenoid to the other pole of the battery, now do you hear a click, you should. if you don't then you have a bad solenoid, you can try this with the new one that is on the mower, just jump a wire from the battery + to the small connection on the solenoid, it should click and turn the engine over. remember to check all the interlock switches, give them some WD or the like.
Make sure that the battery is 100%