Shooting someone's past self using special relativity Suppose A and B are a long distance apart initially. B takes off in a spacecraft in A's direction at a really high speed. Both were aged 0 when B took off. When B about to cross paths with A, A observes him to be 30 years old (while A is 60). At this point, 60 year old A shoots the 30 year old B. B dies.
Now, from B's frame, the bullet has to be fired by A when B is 120. That's because the event of 'bullet firing' must happen after the event of 'A turning 60', and A turns 60 in B's frame only when B is 120. So B has to die at 120 as seen from his own frame of reference.
How can B die at both 30 years old and 120 years old?
 A: Case I:  When B takes off, A and B are both aged 0 in A's frame.  
1)  When B passes A, B is 30 (given in your setup).
2)  But B's clocks run at half-speed according to  A, so A says that B has been traveling 60 years.
3)  Therefore A is 60.  
4)  But B says A's clocks run at half speed, so B says A was born 120 years before the shooting --- that is, 90 years before B was born.
5)  So B's story is this:  "90 years before I was born, A was born.  He aged at halftime, so on my birthday, he was 45.  At that time I started my journey to earth, which took 30 years.  During that time, A aged another 15 years, so he was 60 when we met. Then he shot me.  I died at 30."
Your mistake:  You said that "from B's frame, the bullet has to be fired by A when B is 120".  That's not correct.  The correct statement is "from B's frame, the bullet has to be fired 120 years after A was born".   Since B is 30 at the time of the shooting, A must have been born 90 years before B.
Your bigger mistake:  You assumed that two different problems (namely this one and the one you asked in your last post) have to have exactly the same answer.  In the other problem, A and B were in the same place at the same time when both were born.   In this problem they weren't.  
Case II:  When B takes off, A and B are both aged 0 in B's frame.   
1)  When B reaches earth, he is aged 30 (given in the problem).
2)  According to B, A ages at half-speed.  Therefore A is 15, not 60 as you supposed. The 15-year-old A shoots the 30 year old B.  Game over for B.
A: You suppose that A and B are far apart "initially" and that they were "both aged 0 when B took off". It means that you assume that simultaneity is an absolute notion, independent of the frame of reference, which is why you come to a paradox. It is exactly the symetric case of two people starting at the same point, diverging fast, and observing each other: each one will see the other one aging slower than him, because they compare moments in non parallel time sections. This is best shown by a drawing, this is the one I have suggested in this thread:  link to a video at the second displaying the graph.
A: In addition to Willo's answer
As it was being said by @WillO and @PM 2Ring and others, your problem here is that you have assumed:
'Both were aged 0 when B took off' 
Well that's simply wrong, because like this problem, you have not specified in which frame?! Of course you could have asked
1.When B takes off, A and B are both aged 0 in A's frame. 
or 
2.When B takes off, A and B are both aged 0 in B's frame.
or 
3.before B takes off (where he is at the rest relative to A, but in different position), A and B are both aged 0 in both frame. Indeed, their frame would be same, because they are at rest at that moment. Then instantly, B accelerate toward A and make his velocity constant after a moment in his frame.
But they are different questions. In case 3 You might think that if B zero y/o (after taking off) sends a signal to A, in his frame, he'd assume that A is zero y/o too but that's 100% wrong, in his frame A will be a lot older than B.
Someone might wondering how this asymmetry possible even though their initial state were same, well in short in accelerated frames like B, time can elapse  faster or slower depending on position. i.e in B frame, A's clock will elapse a lot faster than his as long as B is accelerated. And if you assume that B will accelerate instantly, then you will see that in B's frame A ages instantly as well! If you are intested in last case i suggest that you read 
Explorations in Mathematical Physics from Don Koks
Accelerated frame section.
