Will this be available in the NCIC database?
Until Kansas removes it, yes.

Am I likely to be arrested at the border when I attempt to cross?
You may not be arrested, but will certainly be detained long enough for the dispatchers in Kansas to verify the warrant and confirm if Kansas will or won't extradite. If they will, arrest will follow. If they won't, you'll be on your merry way until the next time you encounter law enforcement and you will repeat this over and over until it's cleared up in Kansas and the warrant is removed. If you mean border as in out of the country, this could well be problematic for you as it could be interpreted as trying to leave the country due to the pending warrant.


Will this affect a nexus application?
Possibly

Is it possible to resolve this without having to travel to Kansas and turn myself in?
Not to my knowledge. Kansas has jurisdiction, and the only way that I know of for you to appear before a Kansas court is to do so in Kansas. You could always call the Clerk of the Court and ask if other options exist.