I use the Hop on Hop off bus for the two day pass. It is easy to use. We get to see other sections of Reykjavík of interest outside the old section. https://www.re.is/tour/hop-on-hop-off/
At the stop at Old Harbor:
Saegreifinn (Sea Baron in English) is one of the great restaurants in Europe. Warning before ordering because portion sizes are large...ask what is included with the meal. Our weekday lunch entry included the amazing unlimited salad/tapas bar, desert bar, soup, and salad. That was enough food for days. https://www.saegreifinn.is/
Across the street and at the bus stop also is a fantastic gastro brewery with tours: https://bryggjanbrugghus.is/
Further down the street is Kaffevaginn which was wonderful for coffee and breakfast.
One of the hop on hop off bus stops is at the swim park in Laugadalur. This is a huge swim complex we love visiting. Outside this venue is a hot dog stand worth trying an Icelandic hot dog.
From this park, you are really close to the bayside walkway down to Harpa.
The next stop on the hop on hop off is the Harbor stop. This is where you can rent a car from Enterprise if you need one.
At this stop is the ferry for Viday Island. This ferry ride is only a few minutes long. John Lennon memorial is on this island. http://imaginepeacetower.com/
Definitely stop in Harpa building. Beautiful building (with clean facilities which is few and far between in this city). Go upstairs for great photo ops of the bay.
Hop on hop off stop at Perlan. Perlan is free to go to the roof for a great view overlooking the city. Nice but pricey restaurant. Perlan has a Aurora immersive movie experience that is well worth the price. (Especially if like during our last trip in the fall, every night was cloudy. This was a good chance to feel like we saw the Aurora even though we’ve seen one before.). I didn’t think the rest of the museum was worth the time investment but other folks in our group did the glacier show (you put on crampons to climb a simulated glacier cave) and liked that.
Finally, and this is not in any tourist guide:
There are bike/scooter/pedestrian paths laced throughout the city. These are not only fun to see how the residents navigate sans tourists but are lovely and you’ll see the real Reykjavík. Great short cuts too. There are scooter and bike rentals at most large hotels.
There is a good tourist guide at the airport, be sure to pick up...lists current events going on.
I wished I had this on the first trip. The coupons are worth it.
Wishing you fun and a good trip!