I highly recommend the Best Western Astina Hotel in Shinjuku. It's new, has small but comfy rooms, a good breakfast, and best of all, it's cheap. Single room plus breakfast starts from only $85! It does depend when you go, as the rate is variable, and the day you want to stay (Mondays are cheap; Fridays are pricey). But worth giving the booking engine a whirl to see what deals you can get.
Shinjuku is not Ginza, but you can take the Chuo line across and if you get one of the express trains you can be there in 10 minutes or so.
Here's the link for the hotel: Best Western Shinjuku Astina Hotel Tokyo