Nago-Ji is a Buddhist temple of the Soto School. The temple was destroyed by air raids during the war and the current main hall was completed in 2001 so it doesn't offer the ancient Japanese feel you might be searching for. However, it does have the advantage of offering authentic zen meditation sessions just a ten minute walk from Shinjuku Station.
Zazen is offered the 1st and 3rd Saturday of every month (you can arrive from 1:20p but don't be late, they won't let you in!)
For people with 0 experience in zazen, it's on the second Saturday of every month from 2:00-3:30PM (reserve a spot here)
For people who have been once or more, it's from 2PM - 3:30PM (no reservation required) on the first and third Saturday of every month.
Soto Denjin Boson Nago-ji Temple