I'm not sure if you mean centered above the original window or centered on the screen. Makes a difference, especially because not everyone has their browser at full screen.
At any rate, we've talked about that topic a good bit here. You can uncover even more help with our Site Search (top of screen).
every link of the common window launch in the SAME pop-up window
It's easier than you may think. The standard window.open() method has many parameters, and as long as the second parameter, which is the window name, is the same for each link - then the page will load in that same pop-up window that is already open.