Take the rocker covers off, pull the plugs and turn the engine over slowly by hand.
Watch the valves you should see them open and close.
On each cylinder one should open then close then open a very small amount .
That is the ACR.
Be it a mechanical one or a cam ground one if it is working it will reopen a valve.
Some times you have to put your finger on the end to feel the movement.
We can help you but a small engine apprenticeship takes 3 years so you will have to move well outside of you comfort zone.
Courages generally have mechanical & adjustable valve lashes and unless you have been running without oil the cams usually are OK the singes suffer from broken ACR's, about 1 on 1,0000 .
99.999999999999999999999999999999999999999 % of the hard cranking problems that came in are nothing more than excessive valve lash because they have never been adjusted from new.
So you can go out and buy a new battery, if that makes you feel comfortable
You can go out & buy a new starter solenoid if that makes you feel comfortable
You can go out & buy a new starter if that makes you feel comfortable
But if the valve lash is too big it still won't start because the excessive clearence defeates the decompression.
At lease a dozen come in every year having pissed anything up to $ 500 against the wall cause they were too scared to take the rocker covers off & adjust the valve lash.
Then and only then do you go tearing into the engine.
If it spins the engine really quickly without the plugs in then the starter is USUALLY OK
IF you have been trolling through the archives on this site you will have see people being directed to K & T Mowers we page with a lot the factory service manuals as PDF's on it.
http://www.mymowerparts.com/pdf/Kohler-Service-and-Repair-Manuals/
Go down the the courage twin and download it then read it.