All you’ll need to do is what’s called a passport run and head to Singapore or South Korea (even if only for a single day) to get your Japanese tourist visa renewed.
We stayed in the heart of Shibuya for half a month a few years back at about those same rates. Make sure to drop by an Onsen, the one at the Dormy inn in Asakusa had Fuji apples in the hot room sauna that gave off a really nice aroma. Had the place to myself as well which was also really nice, and there’s a view of the city from the hot tub itself.
