How to improve linoleum backsplash

[Laura DePalma]

My goal is to improve a linoleum backsplash without removing it.  Has anyone successfully primed this product with "gripper" paint or another primer and then painted it successfully. What type of paint looks best (eggshell, semi-gloss, etc) or do you have any other ideas for improving what I have to work with?

A contractor suggested I do this to both linoleum and ceramic tile. I am trying to keep costs down but do not want to see peeling paint in the near term.

Thank you in advance for your advice.

Laura