I recently replaced a HDMI port on a laptop, which would be similar.
I found the hardest part was actually obtaining the connector. In my case it was some weirdo midplane through-hole thing that didn't seem available to the general public. I ended up buying a spare/dead motherboard from overseas and doing a transplant. It's probably worth taking a look to see if you can actually get the connector first.
For the actual soldering process itself, I ended up using a hot air rework gun (Yihua/Atten 858D). It was a bit of a balancing act between melting the solder and melting the connector itself. I found warming up the general area first helped a bit, as well as using a decent sized nozzle. I concentrated most of the heat on the shield pins, as I figured the casing and ground planes would soak up a fair amount of heat.
Having a practice run on the spare motherboard helped as well, so maybe find a scrapped motherboard of any type and try unsoldering/resoldering various parts.