How can I fix this Firewire 800 header by soldering?
It's a little hard to tell from the picture, but the first thing to check for is whether any pads have lifted. If you got lucky, and they didn't, the next step is to clean the pads up - you'll need some flux, a desoldering braid and a regular soldering iron to do that. Finally, it looks unlikely that you'll be able to get to those pins with an iron once it's installed, so you'll need a hot air rework station ($100 or so) to get to them.

I'd also say there's a >50% chance that this won't work at all at best, or you'll break your board at worst.

YMMV.