I'm not a fan of the peel-and-stick backsplash, but if you are, then go for it. As for the tile that's there already, why not just demo it when the kitchen is being demoed, and install fresh drywall on the walls?

There have been several 1920's and '30s kitchen makeovers on this site. Feel free to post your plans if you'd like more input. Good luck!

Thanks for your reply eam44. Here is my situation: I'm 22
years old and this is my first home. Being a home owner at 22, I will
not have the funds necessary to tear out and repair. Being the age I am,
I don't expect to be here more the 5 years at the longest. This is why I
was considering something to just cover right now. I am not demoing the kitchen.

The kitchen is pretty nice looking (until you see the stove/fridge setup which isn't quite ideal). The vintage cabinetry is lovely (although probably not as nice to actually use as it is to look at).

I'd absolutely do something like beadboard. And beadboard would be better in the backsplash than peel and sticks. If you plan to sell in 5 years, peel and stick will reduce the value and overall impression of the house, but beadboard won't.

Invest in a few power tools for your beadboard--either from Harbor Freight if you think it'll be just a few projects, or good ones if you think you'll end up doing more DIY jobs.