oops when I say "pack a picnic", I meant go into one of the convenience stores - Lawsons, 7-11, Familymart - and buy sandwiches, onigiri (rice balls), fried chicken and a few beers. Don't do any work!
I'd go - you say you're not too worried about jet lag, my experience with our son is that he deals with it much better than the grown-ups! And going East is easier than the journey home IMHO.
Parks - an abundance of great choices, my favourites would be Shinjuku Gyoen and Yoyogi Koen. They...