The first thing to say is that the driver must have held a full driving licence for at least one year.
Then, to check the driver's minimum age, please conduct a search and click on the link "terms & conditions" displayed to the right of the car photo of your choice. This will open another smaller window and if you click on "driver's age" you will see what age restrictions there may be and any associated fees. Try a few different car models / suppliers, as they do differ.
If I am 21, will the insurance be more?
At Bahrain Airport the minimum age is 21 for some suppliers/cars i.e. SIXT, so you shouldn't be charged any extra fee as young driver.
Also, AVIS offers some models without any age restrictions...