The micro USB port is part of the lower antenna assembly. If you're handy with a soldering iron or hot air rework station you could replace the port. Just pull the assembly as directed here: Motorola Moto X Lower Antenna Assembly Replacement
Without seeing the actual broken pins and the assembly it's hard to say whether you could repair it or not. Even then, I don't know what your technical skill level is so what I might see as repairable, might be out of your wheel house.