i think you can stay sakura house for around 40-50 dollars a day if you don't mind sharing it with other backpackers. There are also some business hotels which cost the same price, you just need to know where to look.
for transportation, you can actually save if you use day passes like JR Tokunai pass ( 790 yen - more or less 8 dollars) which us can use all over tokyo using JR lines. Very convienient and cheap. Most tourist areas are within the JR lines so you will not have a problem using it.
You can also get the pricey day pass which includes metro subway system.
research on day passes in tokyo
