Take a look at Shibuya Excel Hotel Tokyu. Their "superior double with park view" room is well within your $350/night budget. From what I've heard (I have no interest in staying there, personally) it is a perfectly serviceable 4*(ish) hotel that sits right across the street from Shibuya station.
https://maps.app.goo.gl/xe7V3kHqaoLBSnNW7