Is Rainbow only appear after the rain?

No, Rainbow can also be formed when its misty outside or when there is overspray. You only need water that can act as prism and light to shine to create a rainbow.

Is there a pot of gold at the end of the rainbow?

Having a rainbow is caused by dispersion of light in water droplets which why they occur when sun comes out after rain. And having a pot of gold at the end of the rainbow is just a myth since you can never reach the end of the rainbow.