There are a whole bunch of cracks around the drain in my bathroom sink. Is there any way to repair this or do I have to replace the entire vanity top?
If the sink is integral to the vanity top (seamless cultured marble or solid surface) you'll want to replace the entire top. If the sink is a completely separate piece, then it can be possible to install a different sink. If it's undermounted though, you will need to carefully draw a template of the cutout so that you can get a sink with a shape and size that can work with that cutout.
First, determine what the bowl is made of. If you look underneath and it looks kind of like plaster of paris with random holes, it's cultured marble and is junk. New top and bowl time.
If it's a solid surface bowl, polyester or acrylic, you may be able to remove the sink stopper and aggressively sand out the contaminated thermal shock lines. I've had them all come out, had them reduced considerably, and had no success at all. You never know. Sand with progressively lighter grits and finish with a red or gray Scotchbrite pad.
If it's a solid surface top, the bowl can be cut out and a new one, solid surface or stainless, installed. It may be difficult to make this cost effective, however. It will depend on the size and configuration of your top and your style and budget.