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You have 4 options getting from Akureyri to Reykjavík.
You can fly directly to the Keflavík Airport, or to the Reykjavík domestic airport depending on the time of the day. Time is 45 minutes.
There are busses for you to take. They make stops at gas station/restaurants every now and then. It will take about 6 hours, the same as with a car.
Not the cheapest option. About 6 hours and the are many gas stations, supermarkets and restaurants on the way. Very nice trip.
Car sharing is common in Iceland but it requires you to be very flexible.
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For the flights, as I understand it as long as the overland leg is shorter than the flown legs and the flown legs are on compatible airlines it's normally possible for the two legs of such a trip to be booked on the same ticket with each leg being priced at "half return" pricing.
Google flights offers a "multi-city" option that can be used to look for such flights.
The awkward bit in this case is it looks like Akureyri Airport has few international flights, and the domestic flights from Akureyri to Reykjavík mostly go to a different airport to the international ones, so unless you are very lucky it looks like you will have a connection with change of airport in Reykjavík.
For the car I would expect a surcharge for a one way rental, though I've no idea how much it would be in this particular case.