how can I go to forbidden city from T3 of PEK? From forbidden city to T3 of PEK

Hi, your transit duration is enough to catch your flight. Due to your arrival time, I suggest you take a taxi directly to the Forbidden City if you don't need to have a rest in the airport. You may catch the national flag raising ceremony in the Tiananmen Square. Then have a breakfast nearby and walk to the Forbidden City (it is closed on Mondays).

The Forbidden City is open at 08:30 (Apr. 1- Oct.31) and 08:30 (Nov. 1 - the next Mar. 31). Entrance fee: CNY 40 (Nov. 1 to the next Mar. 31); CNY 60 (Apr. 1 to Oct. 31)

To catch an international flight, you's better get back to PEK at least 2 hours ahead. For the back trip, you can take subway line 1 at Tiananmen East to Guomao. Then walk to Dabeiyao (Guomao Bridge) to take shuttle bus line 1 to PEK directly (the terminal is T3).