Where I live in Alabama, we have blackberries like yours growing wild like weeds on the sides of roads, on fences, in the woods etc. So I'm pretty sure they're a hardy plant, although I've never personally tried growing them because of their weed-like presence where I am.

Considering how huge I've seen them grow in the ground, I'd recommend putting it in a larger pot if possible.

As for trellising, since you're in an apartment, you may want to do something like cutting a tomato cage in half and placing that half in the pot with the plant, so the curve of the half-cage will fit with the pot and it'll be easy to move around. Then just direct the growth of your plant onto the half-cage.

I know this is old but I would like to know what you ended up doing. I also am in the same situation and I'm not sure how to support it in a container.

You could use stakes. It only needs support when bearing fruit.