I would actually suggest you to buy a foam air filter element that is big enough. Usually they are sold for certain specific types of engines, like certain lawnmowers or certain generators. However, you probably are able to buy an air filter element big enough. Then cut it with scissors to very slightly larger than the proper size and compress it in place. Use duct tape to hold it in place, and seal the sides. You probably are missing the air filter cover too, but you don't actually need it. It's perfectly possible to use an air filter without the cover provided that you have some way of holding it in place (in this case, duct tape).
Foam air filters should be gently soaked in oil and compressed inside paper towels to remove excess oil. So there's a danger of getting your hands or clothes oily if you touch the air filter. Usually in real equipment the air filter cover prevents that from happening. But as long as you don't do that, it works well.
Then occasionally you need to clean the air filter in liquid water and detergent, dry it, re-oil it, and remove excess oil again with the paper towel trick. So you need to remove the duct tape occasionally.
Coffee filters don't work. Air filters made out of paper have surface area far larger than the area of the filter itself, because the filter paper is in a steep "V" shape. With coffee filter, this is not possible.
Edit: As for the oil that should be used, any engine oil works. Presumably that's a 2-stroke engine. So you are mixing 2-stroke oil in your gasoline, right? Because if you aren't, the engine will be destroyed in a very short duration of time. The very same 2-stroke oil is probably good for oiling the foam air filter element. There is no special air filter oil, they are oiled with just regular engine oil, either the 4-stroke oil that goes into oil pan or the 2-stroke oil that's mixed with gasoline. Any food/cooking oil will have too short shelf-life and will ruin the air filter element very fast. Any fuel will be flammable, creating a fire hazard.