Start shaft is a 5/16" hex. If you have really big balls and no brains like me, you can start it with a drill & bit. BE WARNED......it's a VERY stupid thing to do.
Spend $250.00 and buy the starter. If that's more than you can afford, no offense intended , sell the Kart. In the long term that will be one of your lower expenses.
Also, If you are as new to Karting as it seems. Run it for a while with that can muffler. Granted, it's slower than one with an open pipe, but so are you. The can is much more forgiving of carb settings and could save you a $500-700 rebuild. As a bonus, you will likely be faster.
As for rebuilding the top end, I would just light it off and play and see what you have. Unless you're seriously racing these motors will last a while. If you are comfortable with it, I would buy a carb rebuild kit and rebuild the Walbro, or a new carb for $75.00. If it has been sitting for a while full of fuel that will cause problems. The big thing I would check is the clutch, you can't ride this thing around the hood without trashing it.
That Kart has some years on it (guessing between mid 90's to mid 2000's) so if you're looking to be at the front of the pack, good luck. But if you want to have a helluva good time on the "cheap" spend a coupl'a grand to get the kart sorted AND buy the proper safety gear, you scored.