i wouldn't even bother oiling it. Just pull the cylinder off and take the piston off and get in there and do it right.
Examine the head and cylinder and piston for cracks.. maybe the piston has a little hole in it?
Ya neva know what you may find.. chances are nothing but lots of carbon and crap gumming up the rings..
The ring groves should be immaculate.. don't scratch or damage the grooves. Google for tips on how to carefully remove and reinstall piston rings.
Don't use force on anything involving rings. On reassembly, forcing the cylinder over the piston may mean you're about to snap a ring. Patience and care are the keys.
a head gasket can be traced and cut from thin aluminum. Pick something close to the original's thickness. Clamp the material between pieces of wood to drill clean bolt holes or use a suitable punch. Or cut everything with a Dremel MotoTool.
No reason to be shy about it .. a moped engine is the perfect engine to start on as far as gaining confidence. The bare minimum of mechanical skills and basic tools are needed.