I am looking for a reasonably priced apartment in Iceland.
I am planning to go late June - late July. All the apartments I have seen on AirB&B seem to be very expensive. Is there a good part of Reykjavik to look in - or should I consider staying in another part of the country altogether.
My plan was to travel there and to find an apartment after getting there, but I read an article the other day saying that all the hotels / hostels get booked up during the summer - so I am little worried now.
You should be able to find some place to stay in or around Reykjavik. By United States standards, it really is a very small city (Population is around 110,000, surrounding area is about 203,000). There are several suburbs that surround the Capital City and as long as you are close to bus lines, it is still convenient to get downtown. I prefer to travel on the shoulder or off seasons so that things are less crowded, but since you are going near the peak season, I'd expect things to be very crowded downtown so the sooner you could book, the better.
Outside of Reykjavik, the only other really built up areas are near the airport in Reykjanesbær (14,000) and a northeastern city of Akureyri (17,000). Other then the Blue Lagoon near the airport, there's not much to do there and you really wouldn't want to spend the entire summer there.
Of course, traveling through the country is an amazing experience and shouldn't be missed, but many of the small towns have less then 1,000 people living there and unless you are really trying to get away from the world, you are far better off near Reykjavik.
Hope this helps!