Bring one cloth that is both water protected and keeps you warm. If you do not find it or find it hassle to carry, bring money and buy from UNIQLO. It will cost you around 7000 yen to buy both waterproof (and warm at the same time) clothes, given that there is a discount going on.
"Wellies" are actually rubber boots, no? The Japanese have actually turned that into another fashion item, the women here wear all sorts of stylish colored rubber boots that are a fashion item just as much as utalitarian. You might want to pick up some local wellies here.