Subway? Too long paths and stairs. For your legs, buses would be the best. JR could be the second. Check this site for bus availabilities.
http://tobus.jp/blsys/navi?VCD=ctop&ECD=srspm&LCD=eI could not judge your pace. Try to put your interested sites in the earlier part of the day. Rough comments on your plan.
DAY 1
JR Yamanote line to Shibuya – see Shibuya Crossing
JR Yamanote line to Harajuku Station
Meiji-jingu Shrine (Taxis could go up to the second parking lot near the Otorii)
http://www.meijijingu.or.jp/img/access/naien.pdfWalk or taxi along Omotesando, dropping at Ukiyo-e Ota Art Museum, until Nezu Museum and garden
Bus(Kuro77)? to Shinjuku Gyoen Garden,
entering from Sendagaya Gate(south) and leaving from Shinjuku Gate(north)
http://fng.or.jp/shinjuku/issue/midokoromap/migoro1609akia.pdfShibjuku WE bus(green) to Tokyo Metropolitan Government (sunset)
http://www.city.shinjuku.lg.jp/content/000197749.pdfDAY 2
Tsukiji Fish Market (after 7am-not auction)
Taxi/Walk to Hama Rikyu Garden
Water bus to Asakusa
Walk(300m) to Kaminarimon Gate
Nakamise Dori
Sensoji Temple
Asakusa Jinja
Nitemmon Gate
Taxi? to Kappabashi-dougugai Dori
Megurin bus
http://www.city.shinjuku.lg.jp/content/000197749.pdfhttp://www.asakusa-e.com/megurin/megurin.htmRickshow
http://www.jidaiya.biz/kanko-j-eng.htmlTaxi to Ryogoku - Sumo museum
Toei Bus to"Kiyosumiteien-mae" for Kiyosumi Garden
DAY 3 (Half day)
Imperial Palace Gardens
East Gardens / Ninomaru / Sannomaru Shozokan/ Honmaru /
Through Kitanomaru-koen Park to Nippon Budokan/Yasukuni-jinja
DAY 4 (Half day)
Hibiya-koen Park
Ginza
Sukiyabashi / Ginza yon-chome ? eg Matsuya
(Suidobashi / Korakuen are the north of Yasukuni and the south of Rikuguien)
Mitsubishi Ichigokan Museum
Tokyo International Forum Building (walk through)
DAY 5
Komagome Station - Rikugien Garden
JR Ueno ?
Ueno Park
Western Art Museum
Tokyo National Museum
Megurin bus or Subway Ginza line for Ueno-Asakusa to Asakusa Pier (around sunset)
Water bus from Asakusa to see Odaiba Bay/Rainbow Bridge
Water buses are not so frequent. Times for Hotaluan & Himiko are not yet determined.
http://www.suijobus.co.jp/en/price/There are slopes around Shibuya, Imeperial Palace, and Ueno Park.