For reasons that are beyond the scope of this conversation, that first night is already booked at the Sheraton in such a way that it's not practical to change/cancel. Would be easier to swallow the extra $200 for the Uber than to book a new hotel for that night.
Oh, I curse it already, don't worry.
But what's trains got to do with that?
A city mini is an oversized stroller in Japan.
I wouldn't waste the time and money to go to Kyoto without staying overnight. If you're okay with a hotel in Tokyo and spending one night away, you can catch a morning train from Shinagawa station and return the next evening.
Otherwise get a simple rail pass like the JR Tokyo Wide Pass (
https://www.jreast.co.jp/e/tokyowidepass/) and day trip Tuesday, Wednesday, and Thursday.
I would recommend Kamakura on Tuesday because it's the closest and you will have a delayed start after getting the pass.
Wednesday and Thursday decide between Nikko, Hakone, and Kobuchizawa. The last one is home to the Suntory Hakushu distillery and is popular than the Yamazaki one near Kyoto. Personally, I think Hakushu is better whisky.
Friday you could hit Tsukiji, Ueno and Asakusa. Shibuya Crossing Motzei Shabbat. Sunday would be Yoyogi Park/Meiji Shrine, Harajuku and the Metropolitan Government Building in Shinjuku. Monday you fly back.