Haneda Airport: For Hamamatsuchō, Shimbashi and Tokyo Tower, take the Tokyo Monorail to Hamamatsuchō: about 20 minutes, roughly ¥520. For Shinagawa/Takanawa, use the Keikyū Airport Line to Shinagawa instead: about 11–15 minutes, roughly ¥330. A taxi is rarely a sensible upgrade unless you have several people and substantial luggage.
Narita International Airport: For Minato’s east side—Shimbashi, Hamamatsuchō, Shiodome—take the Narita Express to Tokyo Station, then JR to Shimbashi/Hamamatsuchō; allow about 75–95 minutes total and roughly ¥3,000–3,500. For Shinagawa and southern Minato, a Narita Express direct to Shinagawa is the cleanest option when its timetable suits, about 70–80 minutes and roughly ¥3,250. Do not choose a taxi: it is dramatically more expensive and traffic-dependent.
Tokyo Station: Tokyo Station is not in Minato, but it is the useful Shinkansen arrival point. JR Yokosuka Line or Yamanote/Keihin-Tōhoku services get you to Shimbashi in about 3–5 minutes; continue to Hamamatsuchō in another few minutes. Budget around ¥150–170 for the local JR hop.
Shinagawa Station: The best rail arrival point for Takanawa, Konan, Azabu and much of southern Minato. It is on the Tōkaidō Shinkansen, JR Yamanote/Keihin-Tōhoku lines and Keikyū line; use the Yamanote or Keihin-Tōhoku northbound for Hamamatsuchō, Shimbashi and Tokyo, or the Keikyū line for Haneda.
Hamamatsuchō / Daimon: This is Minato’s most useful local gateway rather than a long-distance terminal: Hamamatsuchō links the Tokyo Monorail, JR Yamanote and Keihin-Tōhoku lines; adjacent Daimon adds the Toei Asakusa and Ōedo subway lines. It is the right base for Tokyo Tower, Shiba, Takeshiba and waterfront connections.