Many local car rental companies are located within a few minutes walk of KEF airport. If you are travelling light, this may be the quickest option.
Alternatively, it is common for rental companies to send a (free) representative/shuttle to wait for you in the arrivals area. They will be holding a sign with the rental company logo, and a clipboard with names of passengers they are picking up. Even if your flight arrives at 6:00 AM, this service is usually provided.
When you return the rental car, a free shuttle is provided to get you back to KEF airport.
Get the additional insurance. It is likely your existing coverage through your credit card provider will not include things like sand, gravel, and ash protection. This is especially important if you are planning to drive in unpaved areas.
No insurance will cover a broken door hinge. Always open your car door with two hands and a firm grip. Winds in Iceland are no joke! Rental companies will make this one of the first things they check upon your return.
Take pictures of your car and inspect carefully before driving off the lot.
The cheapest 4WD option will typically be the Suzuki Jimny. If you can drive manual, your rental will be much cheaper.
A 2WD will usually suffice if you are visiting in the summer and not doing any off roading.