Is the mount golgotha still exist?

Yes, Golgotha still exists. It is just outside the Lion's gate. In the days of Jesus, the Sheep Gate was at the same place where the Lion's gate is today. (see link)

According to Tradition, Golgotha is under the Church of the Holy Sepulcher. But that Mount was certainly not Golgotha for many reasons. Golgotha was:

1. Outside the city wall.
2. Very near a gate leading into Jerusalem.
3. Large enough for a large amount of People to gather there.
4. Those standing there saw the temple curtain rip.
5. Looked like the top part of a skull (Greek Kranion

   - cranium).