Any old all purpose spray cleaner will be fine. I use Method pink grapefruit....but 409 or Fabulouso....of even a cloth wet with a dish soap solution.
You are not going to hurt the stainless....just don't use a Brillo pad.

Look for a product that is a degreasing agent. Any local cleaning supply company will sell you a product that is made to remove grease. Follow the instructions on the bottle. I usually liberally spray the surface with the degreasing agent and let it dwell for a few minutes. Then use a Dobie pad to scrub it before wiping off the surface with a soft microfiber cloth. In case you're wondering, a Dobie pad is a soft foam pad that is covered with a plastic net mesh (no scratches) making it great for loosening grease. They can be bought at most grocery stores, Target and Walmarts. Hope this helps.

My daughter-in-law used Windex for all her stainless steel stuff.

It seems that what maybe happened is that the oil became a magnet for dust and dirt. Anything that floats in the air will stick to that layer of oil, much like the residue that will settle on surfaces high up in the kitchen.

I just use Windex or a super big car cleaning sponge with Dawn and soap down my stainless finish.

You are going to have to get that oil off of there first.

I use rubbing alcohol too. I don't have SS appliances, but I do use the alcohol on my painted appliances. It cuts grease amazingly well. I think using olive oil might be greasy, perhaps a car wax might keep the SS looking nice instead, without the greasy feel..

Alcohol in a spray bottle, I use on everything on stainless it's great, and what a shine, chrome too and windows and mirrors and on and on... The results will blow your mind!!!

I deal in used stainless steele appliances.The best to clean stainless appliances with is WD40.Try it, you will be amazed.Most people think of WD40 as a lubricant.It was invented for keeping metal clean

Also using a wet magic eraser gets off those rust looking spots that like to form on stainless surfaces without scratching the stainless steele surface

Rubbing anything with any cooking oil would seem to guarantee a collection of dust and grime.