How do I replace my samsung captivate charge port?

It is possible to repair the problem, but with how dense a MicroUSB port is it won't be easy. What you need to do is take the phone apart and remove the motherboard to get to the port. You can see howw here.

Once the phone is apart you'll need a good soldering iron or rework station. You aren't going to able to use the cheapest iron you can get for this repair.

If you use a soldering iron what you want to do is put some flux on the connector and put the braid on the pins. Once you think you have it off just enough lift the wick up while keeping the soldering iron on it. See if you got rid of all the solder. If not, rinse and repeat

Once you do that, go to the ground clips and desolder those the same way. Once you remove all four lift the old connector off.

Once you get it off, take your replacement connector and put it on the board. Then you want to solder the ground pins in to keep it intact and in the same place. After soldering the grounds in, put some solder on the tip of the iron and push down. Do not use a lot of solder. Do this for all of the pins on the new connector. Once this is done verify there are no shorts. If there are, fix them before powering up the phone and charging it.

If you did this correctly you should have a working phone. If you have a hot air station, just mask off the board and remove it that way. It's also a lot easier too.