Anyone got advice on what might be able to grow in a space like this?
We are sandwiched between tall buildings, so it only gets direct sunlight when the sun is directly overhead near noon
We are in Sydney, Aus, so pretty temperate weather, but in the shade these pots never seem to dry out
It looks like the neighbour's bamboo is thriving. That would let you cover the walls in greenery fairly quickly.
I'm not familiar with Sydney's climate but in Europe I'd recommend shade-loving plants like Ivy and Vinca, possibly some Ferns. You might be able to get away with something a bit more exotic though.
If it's warm over night you could consider keeping some house plants outside - at least during the summer. These typically cope much better with the low light conditions indoors. For example Pothos, Monstera, or Spider plants.