How to determine the age of an old galaxy? Recently a galaxy A1689B11 was discovered, having an age of 11 billion years. How that age was determined? News sources reported that some cutting edge technology with gravitational lensing was used. Can anyone explain the exact procedure?
When the article speaks of the "age", what it means is that we are seeing that galaxy as it was a long time ago. We are seeing it as a very young galaxy. The galaxy is very far away, so light has taken a long time to reach us from the galaxy. The light from the galaxy is 11 billion years old. We measure the age of the galaxy by finding its distance from us.

To find the distance we use the Hubble Law, which states that the further away a galaxy is, the faster it moves away from us, due to the expansion of the universe. And we can measure the speed that the galaxy is moving away from us because when an object moves very fast, the light from that object becomes shifted towards longer wavelengths (red-shifted). By measuring light that is known to have a fixed wavelength when not red shifted, we can find the speed of the galaxy accurately, and then use the Hubble Law to find the distance, and hence the age of the galaxy. (source)

A1689B11 has a redshift z=2.54 which corresponds to a light travel time of 11.1 billion years

Gravitational lensing makes the galaxy brighter, so it can be observed at these immense distances, it isn't directly used for finding the distance, but without gravitational lensing, the galaxy wouldn't be visible.