Unfortunately I can't be too much help since I only stayed in Tokyo for three full days during my mid-September Japan trip (and one of those days was mostly taken up by going to the temporary Kirby Cafe haha), but I do have a few suggestions. Not really sure what you like but I'll just say what I enjoyed doing.
Kyu Furukawa Gardens - A small but really nice garden (if you enjoy that sort of thing, which I do haha). Very relaxing to just sit down on a bench and have a snack or something. Unfortunately I went at the wrong time for the roses or really anything else to be in bloom, but it was still nice and the cicadas were neat to listen to. Rikugien Garden is also really close (and more well known I think), but I didn't have time to go to it.
Nakano Broadway - A little mall thing with a whole bunch of nerdy stores. Antique figures, video games, comics, animation cells... that sort of thing. I didn't buy too much there, but it was fun just looking at all the stuff that was there. Also there's a good icecream place there. Can't go wrong with icecream
Oedo Onsen Monagatari - A onsen with a cool vibe. Might be a bit of a touristy trap, but I still had fun haha. Inside looks pretty cool and you get to wander around in a yukata. I couldn't try the actual onsen (ran out of time), but the foot bath where little fish nibble on your feet was pretty cool. Also the train I took to get there had an open front window and so you could see a lot of the city on the way there.
Those are probably the main things I would generally recommend that I've personally been to. A few other things I enjoyed would be the Ghibli Museum (It's not something I would do again and cafe food is bad, MUST get advance tickets), Kabuki-Za Theatre (watching a kabuki play isn't for everyone imo and it's a bit pricey but I liked it), and Peter Rabbit Cafe (food is actually pretty good and I like cute things haha).
As far as other tips...
I know a bit of less basic Japanese things, but honestly 90% of the time I said good morning/afternoon/evening, thanks, yes, and excuse me. I did say things like "looks tasty", "is this amount ok?", and maybe some other minor things but I rarely had to interact with someone who didn't know some English. I did find it helpful to read numbers once or twice (for prices), but most places use roman numerals anyway so that didn't make or break anything. Maybe knowing the man and woman kanji might help (for washrooms), but I can't remember what most washroom signs used right now haha.
I can't help too much with hotels, but when looking for one, try and find one that is close to a train station. That's the easiest way to get around in Tokyo, and probably cheapest as well. And speaking of trains, you'll want to get a Suica, PASSMO, or other similar IC card. It's just a card which you put money on to use the trains. You don't get any discount on the fare (at least I don't think so), but it saves a TON of hassle since you don't have to calculate the fare yourself and buy a ticket every time you go on the train. It doesn't matter which IC card you get, as they all work the same, (personally I got a Suica because of the cute penguin) but just get one for both you and Mrs Jaggo. They also work at some convenience stores (which are actually good to go to if you need a quick snack or drink) and vending machines!
Final suggestion... Unless you really have to or there's some other event that you MUST see, I wouldn't really suggest going in September. It's HOT and muggy! For me it was either go in September or not go at all, but man that weather was miserable. If you're used to that sort of thing then maybe it wouldn't be so bad for you, but I hate the heat. Probably October or November would be better if you went in the fall, but that's just a guess.
Ah sorry if that's a lot to go through!
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