It sounds like the condenser. Its essentially a capacitor that keeps your points from burning up all the time. When they get old and then get hot, they don't hold a charge anymore and rob the power from your spark. This just gives you weak spark, not no spark. Once the bike cools down again, it works well enough to start. But shut the ped off hot and you are out of commission for an hour or so.
Its not too hard to replace. I don't think you need any special tools even.
To get the flywheel off, make sure you have your plug in. Use a ratchet in the square hole in the magneeto nut and hit the end of the ratchet with a hammer in the clockwise motion. *The nut is Reverse threaded* It helps to spin the flywheel till you feel the compression building before using the hammer.
99% of the time this works for me, otherwise I use my impact wrench or piston locking tool.
Once inside, you can replace the condenser and dress the points. The condenser can be either inside under the lighting coil or on the back of the stator. If its on the back, your job will be easy. If its not, you may have a hard time finding a condenser or have to do some drilling and tapping on the stator to fit the round can-style one.
e-mail me if you have any questions.