Where are you flying into?. I would strongly suggest you use the Airport Limousine Bus (orange and white coach) service to take you into Tokyo as taxis from Narita or even Haneda will cost a fair bit. For that reason do not use a taxi.
There are not all that many bargains re hotels in Tokyo. Where are you thinking of staying?.
The metro stations are numbered and announcements on trains are made in English as well. To ensure you're heading in the right direction just look at the station name at the end of the line on the map.
IIRC the gardens of the Emperors Palace are closed on Mondays (as are many of the museums). Do check opening times before setting out.
Do be aware that many Japanese people on the streets do not speak English or if they do its very basic. If you look lost in a station someone will likely try and assist you. There is more of a western influence in the Roppongi are of Tokyo.
Tokyo Skytree is worth a visit on a clear day but you will need to buy tickets on the day of your visit (buying these in advance is only possible if you are a Japanese resident).
Shibuya is full of people and department stores. These tend to open at 10am and stay open till 8.00pm. Do visit the restaurants that are situated on the top floors of many of the department stores because the food there is top notch. Also do see the food halls in the basements of these stores.
Service in department stores is top notch.
Harajuku is worth a visit too but can be particularly crowded on the weekends.
Invest in a decent guidebook along the lines of DK or Lonely Planet.