Can you recommend places for shopping cosmetics at significantly discounted prices in Tokyo and Kyoto?
It's difficult to recommend places for shopping cosmetics at significantly discounted prices. The best quality cosmetics are usually not discounted.
But if you want ok quality and very cheap, you can look for them in Daiso, it's a famous 100 yen shop chain all over Japan that has quality products.
Also, the large department store Donki (Don Quijote http://www.donki.com/en/) has a good variety and sometimes they have sales. You can also bring your passport and buy tax free, which makes it cheaper.