Snap, I've a south facing balcony too - I grow clematis on mine in large pots (2 feet deep, 12 inches wide). I tend to choose ones that need pruning down in winter, so I don't have to look at all the dead growth throughout that season, so choose from the later flowering jackmanii/viticella group, which all need cutting down, so varieties like Comtesse de Bouchaud, Gipsy Queen, Ernest Markham, Huldine. Otherwise, as its southfacing, you may get away with annual climbers such as Eccremocarpus scaber - I find they often survive and keep going for a few years, although I'm in London, don't know where you are, you've not said. Depending what part of the country you're in (its really useful to know that if you want good advice) you could consider Trachleospermum jasminoides, but it may eventually get a bit too large.

What lovely ideas! What about a nice passion flower? Some lovely new hardy ones being introduced to the market these days. I also have a lovely jasmine humile revolutum which has a lovely scented yellow flower and is evergreen. I am growing it in a pot in my front porch and it's doing really well (in the East of Scotland). It would be nice to have something at least semi-evergreen. Rosa 'Veilchenblau' is reliably evergreen, but not sure how it would take to being pot grown. There must be some lovely patio climbing roses though, like Warm Welcome which I have just planted, and they would give you colour, and scent for months.

Thanks for the advice. The apartment is in Colchester and on a river. There are two balconies, one above the other. I was hoping to run the climber from the lower balcony to the top one. I think it would look better with an evergreen but not sure.
Thanks again.

Trachleospermum would probably do that for you - eventually. Lovely plant, fragrant flowers in summer, evergreen leaves.