The Ingólfshöfði reserve is open to visitors on tours, which begin with a fun ride across 6km of shallow tidal lagoon (in a tractor-drawn wagon). After a short but steep sandy climb, there's a 1½-hour guided walk round the headland. The emphasis is on birdwatching, with stunning mountain backdrops to marvel over. Note that puffins usually leave Iceland around mid-August.
Confirm tour times via the website, where you can also book tickets (prebooking recommended). There are often additional tour departures outlined on the site, and there's a link to the Facebook page for updates on what people are seeing on the reserve (important in August, especially, when puffins depart).
Note that there is a longer early-morning tour option specifically for photographers (20,000kr), which must be booked in advance.
The signposted departure point for the Ingólfshöfði tours is close to the petrol pump and booking agency at Fagurhólsmýri. The departure hut is about 2km off the Ring Road; drive down here to reach a small car park.