The open mics I go to have a couple of different systems. Two of them are pre-book, with sometimes a bit of space available for ad hoc arrivals and occasionally a bit of pre-planning on collaborations. The other is book in on arrival and opt for a time slot. Although the pre-book ones have a running order decided by the organisers and you find out about it when one of them says "you're on next", you can ask where you are in the order. The book in on arrival one has the booking sheet with time slots displayed on one of the speaker stands.
When people want me to bass for them, they generally ask me beforehand and tell me which songs (and key/capo) so I can at least scribble notes, although it's not infrequent that I've never heard a song before playing it, which is always fun. Another alternative is looking over their shoulder if they've got charts (not ideal with capos) or (worst option) watching their fingers, which again is sub-optimal when they're using capos.