It sounds as though your rock has already been cleaned and cured which means your job is much easier. It doesnt have to be packed in water for you to transport it, a damp paper towel ontop is all you need or wrap in plastic bag, but your trip is so short, it will be fine.
If nothing else is in your tank, then you can add your rock with no problems and not worry about a brief ammonia spike. Rock does add to the bio load and will cause your tank to cycle to handle the increase, but if your system is up and running and you have no delicate fish to worry about-go for it. YOu will want to check your rock for any undesirable hitchhikers-especially aiptasia, certain crabs and shrimp. Sometimes if you leave itin the bucket or cooler for a while 'things' will crawl out-decide if youwant to keep them or not!
YOur light and water quality will both affect the coraline algae. If your calcium levels, hardness etc are ok, and you have sufficient light, it will only take about a month before you see active growth on your rock-
Now, if the rock has not been cured or cleaned, you will need to do this yourself and it is a tedious process.