Replacing fascia board: what kind of contractor?
I would actually recommend a handyman first.  There are gutters, carpentry, and painting involved and handyman should be adept at the disassembly required, the carpentry, and painting required.

A siding contractor might work as well, but they probably would not paint and may not want to mess with the gutter.

A gutter company would be my last choice as I suspect most would not consider such a repair in their scope.

I would NOT recommend a roofer due to the carpentry and painting.