Possible - Yes, but there are a few things that need to be done first.
First off, you are not able to duplicate a page and its contents within Visio.
The easiest way to do this is to select the items that you want to copy from one page and then paste them on another page. In order to make this work you must first go to the D-Tools Menu and select Options. On the Options Window, select the General Tab at the right. You will see a check box option "Promt to add new product when shapes are pasted into drawing programs", make sure to uncheck this box. See screenshot below
After you have done this you will be able to select the items that you wish to copy to the new page and then paste them on the new sheet. The pasted items will not line up on the exact same location on the new page, however it you keep all the items select you will be able to move them (all together) into a similar position. And if you really want it perfect you can switch back between each page until you get the alignment perfect.