As far as I know, you can get a one DAY pass for the various transit systems but you won't find a one week pass unless you get a JR Rail Pass, which obviously limits you to JR transit.
Here's a one day pass which allows you to use JR and Tokyo Metro :
JR-EAST:Special Tickets - Tokyo Round Tour Ticket (Tokyo Furii Kippu)
As for the dollar shops, there are quite a few but the most popular franchise is probably Daiso. There's one in Harajuku if you decide to go there...
Oh yeah, another good shop for cheap goods is Don Quixote.
Donki Kingdom | Don Quijote Co., Ltd.