1 1/2 days in Iceland. Volcano activity

What time do you arrive and what time is your flight the next day?

Are you flying in on an overnight flight, such as from the US?

That all makes a big difference in what I would make for suggestions.

Thank you I updated it.

You may have issues since this is an extremely last minute thing. Since you want to see puffins, there are puffin tours that go out of Reykjavik, so you could do that on your full day. Otherwise the closest place would be Westmann Islands. If you did that, I would head up there the night of arrival and stay over, see the puffins very early in the morning, do a few things on the island that day and then take the ferry back and stay close to KEF for that night. This is a VERY ambitious schedule though, with no room for error, but it's an option. Otherwise take one of the tours from Reykjavik.

If you do a puffin tour from Reykjavik, then yes the Sky lagoon is perfect for arrival evening. Then the next day you could drive the Golden Circle and some of the south coast before turning back.

For restaurants, it depends on where. In Reykjavik, my all time favorite place is Messinn. Other good ones include the Sea Baron, Sjavargrillid (best lobster tacos!), Islenski Barinn, and the Hlemmur Mathöll Food Hall. Of course the hot dog stand is a must;) In the Golden Circle, Fridheimer is good, as is Mika. You absolutely need reservations at both of these. In Vik, the soup company is very good. Black Crust Pizza is there as well.

That second night in Reykjavik, you may consider going to a local pool. We like Sundhollinn, and have been several times.

I did a South Coast tour that included seeing a lot of puffins by the black beach. It was an all day tour though, from 8-20:00. Not sure that would work for you, but we certainly saw an awful lot in one day and it was a small van, with just 19px and they picked up and dropped off at the hotels. I thought it was worth it as only had 2.5 days there.
If you can go to the SkyLagoon the evening you arrive, that would work.

Thank you everyone.
Ms. Jo.
Where exactly did you see the puffins. We have rented a car so we could just go there on our own.

Assuming that you arrive and depart from the Keflavik airport, why even bother to go to Reykjavik? It's at least 45 minutes from KEF to Reykjavik, and if you are thinking of heading south anyway going to Reykjavik would be out of your way. Spend the night before your flight at a hotel near the airport or in Keflavik town.

The black sand beach at Reynisfjara is a 3 hour drive each way from Reykjavik. There are puffins there. This would be a long day, but it is doable.