If these are simple rips without fraying you may be able to put your iron on patches under the rip and carefully pull the edges together and then iron it so they do not show much at all.

I would be concerned about whether or not the upholstery fabric could tolerate the heat needed to bond the patches.

I would just go with the quick fix: either a sheet cover or zipping up some arm covers.

I had the same issue with my couches and I simply found a piece of fabric that was close in color, hemmed the edges and then used a double sided fusible to iron it directly on to the couch. Did not have any issues with anything and in 3 years that it's been attached have had no pulling away. It blends in enough so that my guests do not see the repair. Cost for the 1/4 yard of fabric probably $2.00.

I would use the fusible to create the patch, but DO NOT try to iron it to your furniture, you might just melt yourself to a bigger mess. Instead of trying to iron it down, I would use Elmer's to glue it, let it dry overnight, and then using a curved upholstery needle, sew it down. You can use a blanket stitch to match or contrast to the furniture depending on the look you want. You can use somple squares, rectangles, or circles, hearts, stars, whatever you like. This is your chance to complement or match the colors in the room. You could toss a few extra patches to make it look intentional.

Or buy some doilies to cover the arms. A vintage look, and you can buy some inexpensive ones. Or check the thrift stores in your area for some.

I bought twin bedspreads for two chairs that I wanted to cover. They were easy to wash and always looked good. The hint of using pipe to tuck in to prevent movement is a great idea. Happy searching for bedspreads.

When my couch was an "early marriage" hand-me-down with worn areas, I used iron-on patches on the arms and they worked great for the years until I could afford new furniture. They fit my budget, looked much better than stuffing peeking out, and didn't have to be fussed with all the time like slipcovers do.

If you have an over-abundance of place mats, use those as arm covers, if you need to secure them, they sell those little button thingies that have a spiral wire on the bottom and you just twist them into the arm of the couch.

I would go to Good will or garage sales and buy a queen size bed spread or quilt to threw over the sofa. Since you don't plan on keeping the sofa I wouldn't spend a lot of money on it. My sister has cats to and she has used quilts and bedspreads on her sofa for a long time. And she has found some very nice quilts and bedspreads and has paid very little for them. And her sofa looks quite nice.