At the center of massive galaxies there is a huge and massive black hole that radiates gamma rays. So at the center of the Milky Way there is also a huge and massive black hole and it also radiates gamma rays. Do these gamma rays reach our solar system? What is the age of black hole in the center of our galaxy? Does it still radiates gamma rays?
Black hole in the centre of our galaxy is not active (which means it is not feeding much matter) as compared to black holes in the centre of other galaxies. So it does not emit that much gamma rays as other supermassive black holes does but still emits some amount of gamma rays. Along with gamma rays cosmic rays are also emitted which are ward off by radiation pressure of our sun's heliosphere.
Firstly, you probably mean that a supermassive black hole (SMBH) is in the centre of our Galaxy, not our solar system. That is indeed correct. The age of the black hole will be approximately the same as the Galactic age. It turns out that the oldest stars in our Galaxy are around 13 billion years old. This means of course that our Galaxy is at least that old. Hence it is safe to assume the central SMBH has the same age.
Very simplistically, we can say gamma rays are emitted from the accretion disk around the SMBH. This happens because of the great speed the particles are swirling around it and the friction that occurs between them. As you know the gamma rays are very energetic photons and have mean free paths that are huge. Therefore they can reach our solar system.