That's what I would do. I'm not sure how thick the wood is, but I would start with a 80 grit sandpaper across the whole top. Depending on how deep those stains are, you might need to take off a decent amount to remove anything visible and you wouldn't want to have a low spot. You could do this by hand with some sandpaper sheets and a sanding block (or a 2x4 chunk), but a random orbital sander would be my tool of choice. I've seen them as low as 35$ at a big box store.

Once I got all the stain sanded away I would give it a wipe down with some bleach mixed with water (max one cup bleach to one gallon water). Then I'd move to 120 grit sandpaper then 220, which would be plenty smooth for me on a bar cart, but you could go higher. I would finish the top with a nice food safe wax or oil, like tung oil or beeswax, they're easy to apply, and maintain. If you plan on placing hot things on the cart I'd avoid the wax though.