Elerad  this really is quite an intriguing story. Sci-fi is not usually my favorite genre to read, but you've lured me in quite well. I'm definitely interested in reading more; please let me know when you post more to this story.
I think the best part of this story is that you jump right in on the action. The first person narrator is describing the action as it happens, and it works. It has the ring of one of the old black and white detective movies from a while back.
The biggest thing for me that didn't work was that you completely lost me two paragraphs from the end when you changed narrators with no warning. I had to re-read that section several times and then go back and check the other time references within the story to figure out what had happened. I'm not exactly sure the best way to fix this. You can take the easy route and throw in a few stars centered between the paragraphs, but I think you would lose the intese flow you have created if you did that. You could show the first narrator seeing the second and make the conversion that way, but it would be difficult. I'm sorry I don't have a better idea for a solution for you. However, as it stands, it is very jarring and took some time to figure out what was happening.
One less nagging issue I had with the story, is that it is getting very close to the base plot of the movie Minority Report. You have already established with the last two paragraphs that your story really is going to go in a different direction, but I just wanted to let you know that the similarity was there for me. As soon as you threw in the background info, rather seamlessly in fact, I started to think of that movie. Just thought you might want to know.
Here are my suggestions:
 The large, dark, somewhat foreboding structure... A long string of adjectives like this right in the beginning in your second sentence is a bit discouraging to me as a reader. Large, and dark are very common words. Also, what is large to me, may be small to you...the words don't really give me a better picture. Forboding is better, but when you use somewhat in front of it, it is basically dicounted. If we have different versions of large, that might effect my understanding of the book. You may have someone jump off the roof and die. Where I'm from a large building is about 5 or 7 stories, but you may mean the building is 25, 50 or even 100 stories. While one is likely to get hurt jumping from one, they are bound to die jumping from the other. Forboding is much better, because it doesn't have to be specific. Forboding might mean to you that it is run down and houses shady people, while to someone else, it might mean something else; either way, the same FEELING, the feeling of foreboding, comes across. Just a thought.
It looked almost exactly the same... This is just a pet peeve of mine, but "almost exactly" is an oxymoron. Almost means nearly, close to, and basically the opposite of exact. While exactly means right on, or the same. When you put the two together, I know what you mean, but the oxymoron just jumps right out at me. You can probably find a more direct way of saying the same thing without using it.
The last sentence of the first paragraph is WONDERFUL...it leaves the reader with a big questions of HOW? until it is explained just a little bit later why. Don't ever change this line! 
I walked up to the front door and pulled it open with the grating squeak of gears that have never tasted the sweet nectar of oil. Another little quirk of mine: the dangling modifier. The way your sentence is worded, it sounds like the narrator squeaked when he opened the door, not that the door squeaked when he opened it. Easy to fix, make it two sentences: I walked up and opened the front door. It gave way with a grating squeak of gears...
I could barely see anything. I would suggest either saying EITHER I could barely see OR I coudln't see anything. As it stands, you are being a bit redundant.
...Greman artist named... This section gives vital information to the story. This is the paragraph I mentioned before that made me think of the movie. The phrase above also made me go "Well, did you?" I really wanted to know if the narrator or someone else had taken care of that particular man, and if so, what were the changes in the history of man because of it? Maybe you plan on introducing this idea later. Maybe that is what the new narrator is for at the end. Just wondering.
The paragraph that starts with People often asked... is full of verb tense contradictions. The tense in this story is going to be hard to maintain as it is. The action that is taking place is in the past, not only for your reader, but for your character; however, since he has time traveled it is the current time for him, and still the past as well. Up to this paragraph you have done a good job of keeping your verb tenses straight. Here I think you should continue to use present tense, though what he is talking about is happening in the future, because in actuality it is happening in his present, though that shifts. Does that makes sense? Let me give you a short example and you can email me if you want clarification.
People often asked me if it ever bothered me: if putting a bullet in a person's head when he or she hasn't yet committed....
People often ask me if it ever bothers me: if putting a bullet in a peron's head when he or she hasn't yet committed...
This reads a little more clearly to me. Do with it as you see fit. 
...and she got into college quite easily. Again, a verb tense issue here. Though her going to college is an event in the past to your narrator, it will be happening in the future to your reader, especially since we only know Rebecca as a child. I would say she would get into just to be more clear.
My gun popped in that intriguing way... I'm not sure that intriguing is the word that you mean to use. The description at the end of the sentence doesn't make me think that the narrator sat back and thought, "Wow what an interesting and questionable way for a gun to "pop" ". I think interesting, plesant, or a number of other words, would do better here.
I am very interested in your story. I would like to see what happens. The new narrator at the end adds an interesting twist to the scene. I think that if you clear up the second narrator's entrace and work on smoe of those tricky time shifts, you could have a great story here. Please let me know when you have posted more, or if you have any questions.
KEEP IT UP!
BlueGrass Girlie
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Being a newbie, I read this for some insight, and not only did I get some, I feel an "AMEN" rising up in my chest.