I hate pantyhose. Japanese ones make me feel like I am hobbled or wearing diapers because the legs aren't long enough, and no matter whether I buy high quality ones or the Y100 store variety, they invariably get a run in them. Mostly, I think it is better to be a woman in Japan except for this one point of being required to wear to nylons to work even on the hottest, most humid days of Japanese summers.
OK, back to sink filters. One pair of ruined pantyhose can make at least 6 sink filters. Just cut those legs up into 20 cm sections and with sections lacking a toe, tie a knot in the end to make a bag. Fit the nylon bag into your sink drain as you would with a filter from the store. Be sure to replace the make-do filter at least twice a week. Since I wear a skirt about once a week, and put a run in my pantyhose after about three wearings, I am able to keep a constant supply of sink filters.
For the bathroom drain (which is also quite a different structure than the ones back home!), you could do the same, and just take it off and throw it out when it gets full, or get a re-usable plastic filter from the dollar store, and wipe the debris off it with a tissue after each shower.