Cat urine is acidic, and if you attempt to clean it up using white distilled vinegar, you may end up with a stinky shag carpet after the fiber dries. Once cat urine dries, it turns into alkaline salts, which vinegar effectively cleans. Yet, you will have better luck cleaning the urine using an enzyme cleaner, which literally eats the urine. If there is soap residue in the carpet, left from a previous cleaning, it must be removed before you apply the enzyme cleaner, as soap will interfere with the enzyme cleaner.
Lay paper towel on the spot to absorb the wet urine; press, don't rub. Remove and discard the paper towel when it becomes wet, and replace with another paper towel. Rake the shag with your fingers, so it is standing straight, and press a section of the strands between two pieces of dry paper towel. Continue until the area is practically dry.
Lift up the carpet, and dry the underside of the carpet and its pad. This may not always be possible.
Sponge the area with water, to remove any soap residue from a previous cleaning. Remove the water in the same way you removed the urine.
Pour club soda over the area and leave for 10 minutes. Remove the club soda in the same way you removed the urine and water.
Place a dry paper towel on the area and weigh down with a heavy object. Allow the area to dry.
Spray the area with an enzyme cleaner designed for removing cat urine. Follow the manufacturer's instructions. Do this the next day, after the spot dries.