Highly speculative (and quite possibly insane) astrophysics question…?

I think that the Master of all Wizards materializes, grabs the other wizard's wand and snaps it in half, saying, "What the |-|ell do you think you are doing by violating the rules of the space-time, continuum? Begone!"

Seriously, since it is all conjecture, I think the ship remains at the same velocity relative to the speed of light in normal space, i.e 0.99 c

"Okay, so as I understand it, in the presence of a cosmic string,"

Sorry, this is garbage science.

"due to the intense gravitational effects, the speed of light is increased in a local manner."

No. If they are dense, the local speed of light is *slowed* as compared to far away. Witness "gravitational time dilation" and "Shapiro time delay".

"This means that you can go faster than our speed of light there."

No, just the opposite.

"So let's say there's a ship flying near a cosmic string, going at 0.99c. There is a wizard aboard. He waves his wand, and magically teleports the ship into normal space. What happens?"

Magic breaks physics. So they could turn into goo, pink unicorns, or still be trapped by the cosmic string.

"Does the ship instantaneously stop or slow down?"

It is your fantasy...

"Is the momentum translated to the equivalent fraction of our c?"

The "laws of physics" do not like magic discontinuities. So you will probably have it speed up to a similar 0.99c away from the string, instantaneously.

"Is it logically impossible to transfer the momentum like that, even if you're magic?"

On no, all sorts of foolishness is possible with magic.

The speed of light is invarient, time dilation applies though.

As magic does not exist you have moved into "pretendy wendy" space, feel free to write your own fairy story.