How can I clean these bottles with skinny rims?  

Try breaking an alka-seltzer in a few pieces and putting that in them with water. Leave over night and rinse them out in the morning.
That works really well. Alternatively you can use a denture cleaning tablet. Just about anything that fizzes will do the trick. You could try baking soda and vinegar but I'm not sure if the smell would be worth it.

If you have a wire clothes hanger, you may try that and a piece of cloth.