Unless that engine has a starter built in, it's not gonna have a way to charge a battery. Usually, on those small engines, the starter is also the generator that makes the voltage to charge a battery.
If your budget is $500, send that engine back and buy a 212 predator and cent clutch. $135-145...... And more than enough power to push you past 35.
For electric start.... Buy the gx/clone electric start kit on ebay. I keep getting a reminder for a guy who sells them for like $125? It's a starter, flywheel, ect.....
EDIT
And if your swift, you can buy dirt bike engines cheap..... The 200cc 5 speed in my kart came out of a dirt bike I got for $30.... Ok, I traded him a 3.5 briggs engine for it.... So $35..... LoL have a 125 dirt bike in my garage that I'm fixing to buy for $75.... It runs too..... I maintain it for the kid owner.....