If I rez Rasmin Bridger after the Runner has passed a piece of ICE, does he get to pay its bit immediatly, or does it take effect only after the following ICE?
I don't know of an official ruling on this, but I believe that they will pay a bit. You can rez Rasmin Bridger "during a run while the Runner approaches a piece of ice and while the Runner passes a piece of ice" (one of the times the rules lists as the Corp being allowed to rez cards) and since the runner is still passing the piece of ice, they will have to pay a bit /after/ passing.
I think it only takes effect on the following ICE. While I also have not seen any official ruling on this card, it hinges on several different rules. When does Rasmin Bridger's ability trigger?
From this chart, step 32, the ICE is passed and this is where the effect triggers if Rasmin Bridger had already been rezzed. If he had not been rezzed, the next opportunity to rez him would be after the runner decides to jack out or not (step 35).
The rulebook only mentions that a Rezzing a Corporation Card can occur (among other times):
The rules don't really clarify the timing of while. The closest thing regarding timing is the Run Protocol - Overview section. It seems to indicate that Upgrades must be rezzed during the approach or just after the runner doesn't jack out.