First you must annotate the schematic. (Give each component a unique reference designator)
In your case i would do this by hand and give the connector the same reference as the old connector had. (This makes importing the changes to pcb_new easier.)
Tools->annotate schematic
Make sure your connector has the correct footprint assigned. (Either start cvpcb to see if all symbols are connected with a footprint, look at each symbol separatly or wait for the error output of the netlis import.)
In a normal workflow you would also run the electrical rules check at this point. (This might give you a lot of errors. It is up to you if you trust the schematic as is or if you want to clean it up.)
Tools->electrical rules checker
After you are happy with the schematic export the netlist
Tools->Generate netlist file
Switch to pcb_new
Import the netlist
Tools->netlist
A few notes for this dialog.
If the new and old connector have the same reference designator, setting the exchange footprints setting to change should do the trick. (The new connector should be placed at the same position as the old connector and the old connector is deleted.)
Be careful with the delete extra footprints setting. This might delete footprint that you don’t want deleted. (Example: mounting holes that are not connected to a symbol, a logo, …)
If the exchange footprints flag does not delete the old connector you can delete it by hand. (Might be the safest option.)
A good beginners tutorial that explains all this in great detail is the getting to blincky video series by @ChrisGammell (Might be of interest for you as an alternative source of information. A video can go into a lot more detail than a bit of text plus a screenshot. I think video 6 has the information i described in this post.)