Dyrhólaey is about 174 km from Reykjavík city or about 2 1/2 hour drive not so far away from the beautiful little village Vík (the southern most village in Iceland). Dyrhólaey is a small peninsula or promontory but is a former island. The whole area is really interesting and I recommend you spend some time there. I personally could stay there for hours. Simply love this place.
When driving on the main road #1 or ring road direction east you take a turn to the right to road 218. Drive that road to an end but please note you have two options i.e. either drive it all the way or make a right drive up the hill. That is because Dyrhólaey is divided into two parts i.e. the higher and lower part. When turning right you are able to drive all the way to top which I highly recommend. When coming down again I also recommend you finish road 218 because on the lower part there is a great view and a beautiful beach to explore. Please also note that during spring/summer time or nesting season Dyrhólaey has limited access. You should check before you leave because it could be different between season. It could be closed somewhere between 1st and 12 of May and then limited access until middle of June. During the nesting season the lower part is open to car traffic but the higher part is then only open to travellers on foot during daytime (09-18). https://guidetoiceland.is/connect-with-locals/jorunnsg/dyrholaey--the-arch-with-the-hole