Normally if a repair depot were tasked with a job like this they would use a tool like
this one to quickly and completely suck all the solder out of all the PCB holes.
Since you only have a soldering gun (and one of the worst in the world probably) you will need to get creative. This is one of those jobs where a Dremel tool's cut-off wheel will really make the job a lot easier. If I was doing this job myself, I would use my Dremel to slice the joystick up into quarters and then slice off the connection pins at the side, one by one. The joystick would fall apart in pieces allowing you to cut it off completely. Then, you go back carefully with a pair of needle nose pliers on one side and your soldering gun on the other, on each of the protruding pins that remain embedded in the board.
Use a
high-quality solder sucker to clear the empty holes of remaining solder. If any stubborn solder won't come out, use a squirt of canned air and your soldering gun together but watch out for loose solder spray landing on and shorting out other components. Then, mount the new joystick, resolder, and voila.
If you like, you can try just using the solder sucker, not bothering with the Dremel. Sometimes you can clear the holes just using it alone but your chances of that on something like this job would be pretty low, I imagine.