Nope didn’t go to any sumo matches…
We didn’t get into the Imperial Gardens for some reason, I think a big portion of it is closed off from tourists. I think Ueno Park may be a good spot for spotting runners and various yoga/meditation activity. Perhaps Yoyogi park as well.
Yeah I remember Shinjuku now, we were there during the evening, pretty cool place. Walked through Kabukicho which is a red light district, known as the world’s safest. We didn’t go into any clubs although a lot of big black males tried to convince us to do so. Just keep walking and ignore them. The barkers (swedish: inkastare) will walk with you but it’s illegal for them to even touch you. When you walk far enough you’ll cross into some other guy’s turf and the first one has to step off so you only have to endure them for 20 seconds 
There is also an area called Golden Gai in Shinjuku that is supposed to have lots of cool small pubs and bars.
This site has some cool articles that aren’t your typical tourist guide fodder: