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Here is what I have:
5091000 Snapper Fast Cut ZT18440KH, 18HP Zero-Turn Rider with 44" Mower, I know this because I used to have a basic manual.
It has a Kohler motor on it. I have to assume it's an 18HP because all of the Kohler Identification that I know to look for is gone.
Here are a few pics:




My issue is electrical. And it has to be something related to the charging system. Here is my reasoning:

Just bought a new battery, (the old one died VERY quickly and I just thought perhaps we left the key on). The new battery starts the mower just fine. Twenty minutes into mowing the motor just shuts off. In this case I was able to fire it up for about 20-30 seconds after turning off the blades and letting it sit for a minute. But then it just shuts off again. Leave it sit for 30 minutes and it will run for 2-3 minutes now.

So, I'm capable of replacing components, but my issues are:
1. I can't figure out EXACTLY which Kohler motor to buy components for since my Kohler ID tag is gone.
2. I am unsure of which electrical components I should replace, and in what order?
3. Does anyone know of some good videos or tutorials on how to replace these components?

Hope y'all can help out!

Thanks!

The thing poking out of the blower housing with a 3 wire plug on it and another ground strap to a mounting bolt is the rectifier regulator.
It controls the charging of the battery.
The two outside wires are the AC, measure them with the engine running, should be somewhere between 30 & 40 V AC
Then measure the purple wire to a good ground purple wire to the battery - and the purple wire to the ground wire on the rectifier, all should be the same and all should be 13.0 V DC to 14.5 V DC.

There was a problem with the purple wire running backwards through the ignition switch overloading the contacts in the ignition switch so on some mowers they moved it directly to the battery.
Because they are exposed ZTR engine suffer badly from corrosion so check all of you terminals and cables, BOTH ENDS as you get corrosion under the ground lead which no one ever checks.
On ZTR's I always paint over the ground connections with "liquid Electrical Tape" to prevent this happening.

Thanks Bert!

I will check it out tomorrow.

If it comes back as NOT GOOD, how do I figure out what Kohler rectifier/regulator to purchase?

There is only 2 a big one & a small one.
Your motor is a Magnum series .
Most mower shops keep the rectifier as it fits nearly all Kohlers with electric starts & electric PTO's
But you are getting ahead of yourselfusually it is nothing more than a broken wire or bad connection.
They are a very reliable unit.