I was playing for the first time last night with the Jinteki corp and i noticed that some of the ice have subroutines that say "end run" while others don't. My question is if I have only a "Neural Katana" guarding a server (3 strength and has 1 subroutine that does 3 net damage) does the runner get to pass it regardless of if he breaks the subroutine or not?
What stops a run? Some ice has end run sub-routines, some don't
The run ends when the runner chooses to end it or they dont break a subroutine that says to end the run. In your example the runner would keep going and encounter any other ice/access cards or choose to end the run.
Three things can end the run:
1) The words 'end the run'
2) The Runner voluntarily Jacking Out (at a legal to do so time)
3) Flat lining the Runner
In all other cases, if the Runner if still alive after the effect of an unbroken subroutine is executed, the Runner may keep going.