Iceland is expensive and Akureyri is no exception, but a good date here does not have to clear 20,000 ISK. The town is small enough that walking is free, the public infrastructure is genuinely good, and several of the best things to do cost almost nothing.
The pool date: about 2,800 ISK for two
Sundlaug Akureyrar charges around 1,400 ISK for an adult, and once you are in, you can stay for hours. Bring your own towel to skip the rental fee. The hot pots, the steam room, the slides if you are feeling silly, and the long soak in the big outdoor pool with the view of the mountains. This is the single best value date in town and it is not close. Total for two: under 3,000 ISK if neither of you needs a swimsuit rental.
Botanical garden plus coffee: about 2,200 ISK for two
Lystigarðurinn is free. It is one of the northernmost botanical gardens in the world, it is beautifully kept, and from June through September it is open until 10pm. Walk it slowly, sit on a bench, then walk down to Berlin or Kaffi Ilmur for a single coffee each (around 600 ISK each) and split a piece of cake. You have just spent two hours together for the price of a sandwich.
Bookshop date at Eymundsson: 0 to 2,000 ISK
The Eymundsson on Hafnarstræti has a cafe upstairs and a generous attitude about people sitting and reading without buying. Pick a book for each other, swap, read for 20 minutes, talk about it. Buy one secondhand book between you on the way out if you want. This sounds twee on paper and works surprisingly well in practice.
The harbor walk and a hot dog: about 1,400 ISK for two
From Ráðhústorg, walk down to the harbor and follow Strandgata north past Bryggjan Brugghús and out toward Sandgerðisbót. The path runs along the water, the views across the fjord toward Vaðlaheiði are excellent in any weather, and at the end you can grab a pylsa from one of the gas station stands for around 600 ISK each. Two hot dogs, two cans of Appelsín, total under 2,000 ISK. This is unironically one of the better dates in town.
Cinema at Sambíó Glerártorg: about 4,000 ISK for two
A ticket at the multiplex in Glerártorg runs around 1,900 ISK, and the cinema itself is nothing special, but it is a real movie date for under 4,000 ISK total without snacks. Pair it with a cheap dinner at Greifinn next door (split a large pizza for around 3,500 ISK) and you have a full evening for under 8,000 ISK.
Akureyri Art Museum on Kaupvangsstræti: about 3,600 ISK for two
Admission is around 1,800 ISK per person, the building is small enough that you can do it in an hour, and the cafe at Hof a block away is a perfect post-museum debrief. The exhibitions rotate often, so even if you have both been before, there is usually something new. Good rainy-afternoon move.
Free moves worth knowing about
The steps up to Akureyrarkirkja and the view from the top cost nothing. The Christmas Garden, Jólagarðurinn, on the road south of town is free to wander even if you do not buy anything, and it is genuinely strange and charming year-round. The summer concerts in the square at Ráðhústorg during the Akureyrarvaka festival in August are free. The walk along the river Glerá from Glerárhverfi up toward the small waterfalls takes about 40 minutes round trip and most tourists never find it.
The trick to budget dating in Akureyri is that the expensive options (the fancy restaurants, the boat tours, the spa drives) are not actually the better dates. The pool, the garden, and the harbor walk are what locals do, and they are what visitors should do too.