Days 2 and 3 may be hectic, depending on your destinations and crowdedness there.
What would you visit in Shinjuku, Shibuya and Roppongi areas?
You plan to visit the East Gardens of the Imperial Palace, right?
In order to visit the Imperial Palace itself, you need to apply for a tour organized by Imperial Household Agency and get a permission.
Regarding the order of places, you can visit
Tsukiji Wholesale Market (near Tsukijishijo Station (E-18): Exit A1),
Edo-Tokyo Museum (near Ryogoku Station (E-12): Exit A4),
Koishikawa Korakuen (near Iidabashi Station (E-06): Exit C3)
and Tokyo Metropolitan Government Building No. 1 (North / South Observatory) (near Tochomae Station (E-28): Exit A4)
in this order, using Toei subway Oedo Line counterclockwise from and to Shiodome Station (E-19) just near your hotel.
If you can choose one day when none of the facilities is closed, they sound enough for a medium-pace traveler to visit in a day.
You might add Zojoji Temple and Tokyo Tower (near Akabanebashi Station (E-21): Akabanebashi Exit) at the end if you are a high-pace traveler and you start in early morning.
Concerning Tokyo Tower / Tokyo Skytree, if you plan to climb it with a same-day ticket, I recommend you would allot a weekday to it, considering possible crowdedness.
Please note that you cannot purchase in advance an admission ticket of the upper observatory of Tokyo Tower / Tokyo Skytree. Only visitors in the lower observatory can be admitted to the upper observatory.
By the way, are you a good walker?
Omotesando Street, which is manly on a slope, is not yet listed. It appears in the japan-guide.com article featuring Harajuku.
- Harajuku
http://www.japan-guide.com/e/e3006.htmlMeiji Jingu Shrinefs main building and Harajuku Station are located a little apart.
You may hope to take one train on Toei Subway Oedo Line from Shiodome to Yoyogi, walk to Meiji Jingu Shrine and take the route below to Harajuku.
Meiji Jingu Shrine: Yoyogi Gate [North Gate] --(North Approach)-- Main Building --(South Approach)-- Harajuku Gate [South Gate].
http://www.meijijingu.or.jp/english/map/2.htmlFrom Harajuku Gate of Meiji Jingu Shrine, you can proceed to Omotesando Street.
If you plan to visit Takeshita-dori Street, you can make a detour by the street on the way back to Harajuku.
From Harajuku Station you can get to Shibuya Station by JR on one train.
An area which can be called Shinjuku is very wide, including districts of Shinjuku and Shibuya as municipalities.
TMG Building and Meiji Jingu Shrine are located away from Shinjuku Gyoen National Garden, which has gates in Shinjuku and Shibuya city wards.
The Shinjuku railroad terminal is composed of several stations on many lines, but they are not always the best stations for your destinations.
Shibuya Stations (of four operators on many lines) are at the bottom of various slopes. Redevelopment works have been ongoing in and around Shibuya Stations.
When you tour Shibuya, I recommend you would rely never on guidebooks issued in 2012 or before but on newer well-updated sources.