I just bought a '97 Gp1200, and exactly like you said.. takes a second to start it up, but once it's running, runs great. I have the primer installed on the side, So when I first launch it or it's been sitting I do prime it a little and it helps, but sometimes you start floating out trying to start it and it sucks lol. If you don't have the primer, maybe it could help you out as well.
I think this is the kit the previous owner installed,
RIVA Online Watercraft Performance Catalog