You don't say how wide or deep the hole is.
Let me guess.
It is 20mm square and 3mm deep and the wall is emulsion-painted in off-white.
You can buy a tube of Fine Filler at your local DIY or hardware store.
Also buy a broad-bladed filling knife whose blade is at least 50% wider then the hole.
Pick up some colour sample cards for emulsion paint.
Read the instructions twice.
Read the instructions again.
Clean any dust out of the hole.
Use the knife to press the filler into the hole. By pressing against the solid plaster around the hole you can press it flat. It will tend to bulge out slightly after the knife passes.
Don't fiddle with it.
Clean the knife thoroughly using a pan-scourer before it sets.
When set firm, you can use the knife as a scraper to scrape off the slight bulge. You may also, if it is not fully hard, be able to moisten the surface and polish it smooth with the knife held almost flat to the wall.
You will now have a white patch on the wall.
Compare the colour samples to the wall. In a rented flat it is very likely to be magnolia, or almond white, or barley white, or rose white or one of those other shades. Decide if it is matt or silk.
Buy the smallest pot you can find of the appropriate colour. It might be available in a tiny "trial" pot. Apply it lightly with a sponge. You will find that the colour does not match perfectly because the wall is dirty and the paint is clean.
The day after applying the paint, when it is fairly dry, sponge the walls where dirty with a little washing-up liquid in clean water. Dab this dirt over your paint patch to make the dirt blend in. You can do the same with other paint chips you have made around the flat.