If you put all of the duplicated circuitry on its own schematic sheet, and place multiples of that sheet as subsheets in your schematic, you can have Altium generate rooms for those duplicated sheets. Then once you have one room laid out, you can copy the layout and routing to the other rooms. This works quite well when the duplicated section is fairly complicated and you have the space on the board to have all of the sections laid out identically, but it can be a bit of a pain if you need to have slight variations in layout. Also, the designators tend to get unwieldy because of the way Altium handles annotation of duplicated sheets.
If I've got a fairly small bit of layout I want to duplicate a couple of times, I usually just filter the tracks+vias+polygons, copy and paste, and then snap component pads to the ends of tracks to get the parts placed correctly. Then D,N,U to fix the net assignments, or even just delete and re-paste the routing copper and let it pick up the right nets from the component pads.
If a lot of your nets are the same from one section to the next, you can also "Paste Special" (Ctrl+Alt+V) and check 'keep net names'.
For manually correcting net associations, there are several useful tools under Edit->Select, E,S,P: Select connected copper, E,S,C: select connection, E,S,L: Select connection on single layer.