Been a couple of times, plenty of accommodation in Tokyo for £100 or less, I like the APA chain, small but functional modern rooms. In fact quite a lot of hotels in Tokyo have small rooms.
I stayed in Shinjuku and Asakusa, both were decent but you have more going on in Shinjuku. Akihabara or gaget alley as you refer it is worth a stroll, would suggest staying until it goes dark as all the street is lit us with the famous signs.
Cant recommend going to Kyoto for a few days enough, the nicest place I visited in Japan with a lot more ‘old school’ Japanese culture evident and more old buildings etc.