Is there a non-surgical procedure to relieve a brain aneurysm?

Yes: There is a very clever technique useful in many but not all cases in which a coil is threaded through a catheter under radiological guidance by an interventive neuroradiologist, releasing the coil into the aneurysm which then clots and eliminates the risk of the aneurysm. So no surgery is involved except the incision to introduce the catheter.

Usually none: There are two main ways to treat them: 1. Coiling them 2. Clipping them. See your friendly neighborhood neurosurgeon to see which is best for you.