I'm sure youtube has a tutorial on this sort of thing. Basically if I have charred/burned board I usually scrape away the burned stuff with a blade or metal brush, common soldering tools experienced guys may have, but you can get the job done with any number of common tools. Next, install you parts, solder up the parts where the traces remain, then use some small gauge wire (solid core makes things easier) to solder a connection where the burned up traces would have gone. Cat5 or phone wire make good choices as they are thin and usually solid core. Try to cut the wires to precise lengths so they sit neatly on the board so they wont get disturbed when handling it.
The most simple solution would be to remove the transistor and the resistor and solder a wire directly from the ATMega SVO pin to the SERVO pin on the CAT5 jack. The servo will work without the servo booster which is what that transistor or resistor make up, though it does help stabilize the servo over longer cable runs. I personally would get the booster back in there, as it is a very simple circuit, easy to hand wire or repair.