I've not been to the job site so I don't know if there's any hazards (downed lines, unstable roads, etc.) that exist. But generally when there's a bridge out local traffic is permitted to travel up right almost to the bridge if they need to access something on the detoured road.

I'd say go for it and if you see a flagger, cop, or foreman just give a quick shout out your window asking if you can proceed safely and they'll let you know what to do. But obviously there may be other issues beyond what a typical bridge site has that may make this different. Whatever you do, don't approach the culvert area; the embankments by the culvert are almost certainly unstable.

And don't pass any signs or persons that indicate danger ahead. They may also need to close down the area for work to stabilize the site so you might be stopped short of your destination if they need a large area for equipment or what not; but I would hope that worst case is you likely could park close to your appointment and walk to it.

Edit: The proper signage you would normally see is they'll have a detour sign and some barricades with a zig-zag path between them and another sign that says "local traffic only". Then you're allowed to pass. But, again, this was an unplanned detour so I have no idea what signage they've managed to put up.