Home Improvement Questions? Ask a Handyman for Answers
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First pic just looks like the stained finish is worn. The fix is you find a stain that matches then rub the stain on just the worn section as much as possible then wipe off the excess Then use paste wax to finish. The second pic I can't tell if it's string inlay or painted lines. If it's paint anything you try is likely to make it worse. If it's string inlay then use Formby's build up remover to clean the piece.
It looks like string inlay. Unless you're a skilled furniture restorer this is beyond your ability. You're best option is to clean the piece which should make the inlay stand out more.
That's the safest approach. Keep in mind with antiques you want to preserve as much of the original finish as possible. But in this case I'd try cleaning not refinishing which would remove the original finish which looks pretty good except for the areas where the original satin has worn off. Replacing string inlay is an extemely skilled task. Unless you're willing to pay a pro just cleaning this piece is best. Any unskilled attempt at inlay repair would make it worse and detract from the value of the piece.