Hi Ben,
You might also want to look at Redcliffs and Mount Pleasant which are areas near to Sumner on the City side, or perhaps Scarborough / Taylors Mistake which are on and over the hill overlooking Sumner.
I suggest you get hold of The Press when in Christchurch - Wed's and Saturdays are best for accommodation/rentals.
In Sumner you are looking around $350 for a 2/3 bed house and VERY rarely available are apartments for approx $300/320. Grab a paper, make a schedule of viewings then just spend the day driving around - you'll need to see a few I can guarantee that - and read this forum carefully so you know what to expect from NZ rentals.
Redcliffs/Mt Pleasant are around the same price. North Brighton will be cheaper.
Do you have anywhere to stay while you look around? If not, Kiwi may be able to help, perhaps drop her a PM?
Hope that helps.
Moorf