What are some cheap options for soundproofing a wall (can’t be permanent as it’s in a rental) that can also be styled to blend in with decor?

Alan Chenkin, Carpenter and “how to do it” guy. Making dreams happen with tools….
Here is a relatively inexpensive suggestion:
Make frames out of 1×2 pine, cover them with a simple fabric, like burlap and then glue/fasten foam sheets (even from the packing that items are delivered in, they can trap sound). Then you can either hang or lean them on the wall, and that should reduce some of the sound.
If you need even more soundproofing, consider Acoustic Panels Studio Foam Wedges 1″ X 12″ X 12″ Which can be stapled or fastened to the frames with 3M Foam & Fabric Spray Adhesive.  These foam panels are used in recording studios to cut background noise, and trap noise effectively.
You could also try these Amazon.com: soundproof wall panels , and fasten them with an adhesive that can be moved (Note: this can leave marks on the wall, but may be cleanable. use your judgement) Available on Amazon.com: spray adhesive repositionable.
Hоw to Sоundрrооf Anything You Can Think Of! Get Rid Of Noise And Enjoy The Silence
Best of luck with your noise control project!