The rain would only affect the rooster if it encouraged air born mold spores.
He probably has a sinus infection or a mild respiratory illness. It may resolve on its own or you can go ahead and treat it with Tylan 50 if you can locate a place to get it and if it's legal where you are to buy it.
If you can't get Tylan, if you have any relatives or friends that have recently had cataract surgery and have leftover eye meds, those can be used in the roosters eyes and it will treat the sinus infection. I've done this and these meds work very well on chickens.
If this has just cropped up since the wet weather, it could be a sinus infection caused by mold. That would be more difficult to treat.
Your location is important, if not critical. Depending on where in the world you live, it's either easy to find these meds or impossible or illegal.