Around Queenstown the best crag by a country mile is Wye creek. Some of the rock is excellent and quite interesing to climb on and there is a lot to do in the high teen- mid 20 range. You can camp for free in the paddock just inside the gate but would need a tent, the crag starts about 15min walk up the hill with the best area accessed by walking across a waterfall then on top of a aqueduct. Great views over Lake Wakitipu. Other good stuff in the area is the ridge traverse of the Remarkables or a jaunt up Single Cone, though very weather dependent and can be very alpine even in January (ie iceaxe +/- full mountaineering gear needed). Some multipitch rock routes
For 1-2 day walks, avoiding the crowded huts on the "great walks":
from the Remarkable ski field across the saddle and down into Wye Creek valley. This should take 2 comfortable days and there are bivy caves, or you can just walk up from Wye Creek and return the same way.
There are also some interesting walks around Arrowtown that follow old mining roads and visit remote mining ruins that you can do overnight
Apparently the day walk to Ben Lomond is also quite good. (up behind Queenstown)
From Kingston you can access the range above, seen south from Queenstown over the lake (Mt Cedric) and there is also some easyish rock climbing at Kingston which is most memorable for the views.
Queenstown Rock and Ice is a good reference for climbing.
Don't waste your time going to any of the crags near town. They really are pretty bad, they make Mt Iron at Wanaka look good!
Around Wanaka there is lots of good climbing, especially out beyond Gleddhu Bay.
A good overnight walk is to walk all the way to Scots bivy from Raspberry Flat and return the next day. Alternatively you can go to Liverpool Bivy (a small hut) or French Ridge Hut easily in a day if you are not laden down with mountaineering stuff.