How to fix wood veneer night stand?

Duffy (cute name), I would use sandpaper in my hand and a sharp chisel to get the veneer off. Two years is not long for a surface to last. But since your veneer has both stains and bumps, it’s no konger usable. The layer under that may be usable. Just sand, stain/paint, seal. Or you can find or buy a new top for the table. Sand all the edges of the new top, glue it on with wood glue or Gorilla Glue or Elmer’s Strong Glue. Center it carefully. Then sand the top, stain or paint it as you wish, and seal with polyurethane. You could paint with chalk paint after you have sanded the top and any more of the table that needs it. You might want to paint with Unicorn SPiT for a really unique top. Use poly for that, also, at least three coats. Use coasters for all drinks and clean up spills quickly. And no fingernail polish or cigarettes. You can make this table better than new. Best wishes ☺️

In addition Duffy, to what Jewell said, you may also want to after taking off the veneers just make your own Salt wash paint:

https://www.hometalk.com/diy/decorate/walls/create-your-own-salt-wash-paint-32241608?expand_all_questions=1

Or

http://www.bluegrassjunkrevival.com/index.php/2016/10/26/how-to-create-your-own-diy-salt-wash/

For painting the sides and use a decent Vinyl Contact Paper on the top:

https://www.google.com/search?q=Vinyl+Contact+Paper

Vinyl contact paper more withstands the everyday living of Condensations from Coasters [or repurposed Compact discs, with saltwash paint same, which will always match the saltwashed furnitures, and also assist consuming the extra paint with a constructive project]

Cover the tops with an attractive fabric table runner.

-Fast, easy, with no messy sanding, or special tools needed.