The soap dispenser will have a nut underneath the countertop that you will have to remove if replacing the entire unit with a new one. this is usually the best option. If replacing just the bottle and pump, check your supply store, they should be readily available. Bring in the old pieces.
To remove the cartridge, you'll need to unscrew that ring with the notches at 12, 3, 6 and 9 o'clock counterclockwise. That usually requires a spanner wrench easily obtained from your home store or plumbing supply store. Make an imprint on paper to get the right size. You might be able to use a slotted screwdriver at one of the notches and tap it with a hammer to loosen it up, then the cartridge pulls straight up.
The wrench will look something like this one for Delta faucets from Amazon. You'll have to check for one at your home store that fits your valve. I have made some by grinding out the excess material from an appropriate sized piece of metal conduit.