GPS

AA batterys can be swapped when flat. Much quicker and more reliable than an internal battery and power pac

Definitely with @geeves on the AA batteries. Just make sure you use Lithium batteries as "normal" ones will run down in about 8 hours on most models. Whereas lithium last about 40 in my experience. I use an Etrex 20 from Garmin. It cost about $260 though that was 6 or 7 years ago now. Also came with a deal where you could send away for the topomap SD card for free which was a good deal as the SD card was normally about $240 from memory. Not sure if they still offer that deal. Even without I imagine you should be able to get the unit+maps for around your $500 budget though you may have to wait until it comes on special.

I have an etrex 30 bought new with colour NZ maps for around $300 three years ago. I run rechargeable batteries on it. On the trips where the GPS is on for all-day tramping, it will easily run for 2 days, if not longer. Take a spare set and your GPS will most probably run for 4 days or more. Try to get the highest mAh batteries you can though.

I have a Garmin Oregon 600/650 I'm not using anymore if you're still looking. It has the NZ Topo and the, what I consider better from Mr Purple (free). Let me know if you're interested and make an offer.