Do you have a printed electrical schematic of the mower? Do you own a volt/ohm meter? I don't know how tech savvy you are but if you don't have these two things its going to be very hard to repair your mower. You also don't say if you are trying to make/repair your wiring harness or replace it with a new one.
Basically you need to determine what each post in your replacement switch does. One post on the switch should make contact with either another post (to ground) or the switch housing itself when the switch is in the off position. Same post shoud not connect to any other post or housing when switch is in run or start position. That post should connect to your magneto.
You need power to the fuel solenoid (if it has one) when switch is in run & start positions. You also need connection to the alternator when the switch is in run and start positions.
Some alternator circuits go directly to the battery, but the better ones are controlled by the igntion switch.
you need power to the start circuit when the switch is in start positon only. There will probably be some safety interlock switches in this circuit as well. I don't recommend bypassing any.
Some units use interlock switches to find a ground for the solenoid energize circuit, this is why an electrical schematic is so helpful.
Take this for what its worth. If you don't feel up to it, best let someone work on it who knows what they are doing.