I own both of these games myself and I have to say that their both amazing. However, even though Uncharted 3 is extremely entertaining and the graphics are so technologically advanced for our console period, it's story will not last you long. I've beaten the game on Charted! mode easily and right now I'm just looking for treasures to complete the trophies. I would say rent Uncharted 3 because it will last you about a few weeks at most unless you're hooked on the multiplayer. Buy Arkham City...many hours of gameplay and challenges/side missions to keep you busy well into the upcoming months and dislocating some bad guys' elbow is always fun...especially if you're the Batman!
Nobody can give you an exact date, but have others have said it generally takes about 3-5 days or so depending on:
A: The availability of the game. If you try renting a newly released game that is very popular and you know many people are dieing to play this game, chances are that you're going to be waiting a while so that adds to the time you're waiting to receive your game. I tend to stay away from games with generally Low availability until they rise to about Medium or higher.
B: Your location and how close you are to your primary shipping center. Like, I live in North Carolina and mine is in Pittsburgh so it takes a few days to get here. But, say that I lived in or very close to Pittsburgh; I would receive my games much more quicker than others.
C: Your game plan. If you are signed up for one game at a time, you have to wait for them to receive your previous game before they even try to ship another. If you are signed up for 2+, same concept but it gives you more options.
I hope that I helped.
I'm fifteen so I'm not completely sure yet, but I'm thinking it will be a video game designer and/or something to do with just computers in general. Video games are my passion in life and it's what I know best, so why not have a job doing what you love? Rather than being a miserable desk-bound employee at some low-paying company my whole life. I'm also really interested in music; singing in particular. So that's always a possibility.
They're all amazing games and the series is definately worth a playthrough. I haven't played much of the first two, but I remember that they were really great. I'm currently playing Human Revolution, and I highly recommend buying or at least renting this game, as it is just amazing. It's a perfect mix of RPG and FPS into one well-designed game with a great story, stunning graphics, and many hours of gameplay. If I had to compare it to a few games, I would compare it to Mass Effect, F.E.A.R, Metal Gear Solid series, and I can somewhat see a little Fallout 3/New Vegas included in there (you'll know what I mean if you play it :P). But, yeah you're missing out if you don't at least give these games a try. Hope I helped. :)