Take a walk through Ueno Park (museums there to see as well)
Take a walk through Yoyogi Park (nice and shady on a hot day) and visit the Meiji shrine
I know you said you don't like shopping but if you like music at all you should visit the huge Tower Records store in Shibuya, plenty of listening station to check out all kinds of music. You could also visit the Tokyu Hands store in the same neighborhood, plenty of everyday but interesting stuff to look at.
Go to Yokohama and walk around the Minato Mirai area.
I did all of the above a week ago yesterday. Tokyo is not boring. Ditch your friend and go exploring, be a little adventurous.