If you really want to try a climbing rose, I would try growing a dark ivy on the trellis/fence this year to create an evergreen backdrop that you can intertwine the rose in next year so you have all year interest and it won't matter if the rose fails! My neighbour grows substantial climbing roses from the basement to the first floor balcony but they are planted in barrels and intertwined with wisteria.
If you paint the fence and trellis a colour like dark grey, it will fade in the foliage and make the space look larger, and hide the dirt, although white will make it feel bright, wood will feel rustic.
I would probably try to beautify the fence. You could put up jali fretwork panels, which will double as trellis. Even just one panel as a focal point would look good.
www.screenwithenvy.co.uk/collections/all-garden-screens
I really rate the Ikea solar lights for longevity. Good lighting will make a huge difference to being able to enjoy the space in the evening and give you a pretty view from the windows at night. I also like outdoor rugs on roof terraces to make it feel more like an outside room.
Are you in London? Colombia Road Flower Market is open again. You should be able to pick up 12 trailing and upright geraniums for a tenner at the end of the day when they reduce prices, maybe some lavender too. Easy to grow, drought resistant, evergreen, they flower forever and are very Mediterranean. I would use the geraniums in balcony/window boxes on the fence and in the hanging baskets. I would start with ivy, maybe a passionflower and some clematis (one winter spring, one summer) as climbers. Taylors clematis are a good nursery. I doubt you will find a large jasmine this year so leave space!
www.taylorsclematis.co.uk/clematis-may-to-september/?filter=allinstock
I would then add in some herbs and architectural evergreen perennials to create a year round backbone for the space eg a small olive, cyprus or citrus (you may have to take it in in the winter), box balls, palms, grasses, a Fatsia japonica (in a shady spot), cordylines (Torbay palms), Trachelospermum jasminoides (scented), rosemary and bay (standard or pyramid can look good). Hibiscus is good too but not evergreen. Oleanders are easy and drought resistant but very toxic if you have pets/DC. Nicotiana sylvestris is also easy to grow flower and strongly fragranced. Nasturtiums are another easy to grow, drought resistant flower that will trail or climb if you want to fill space quickly this year.
You can divide up supermarket herbs to grow on. Norfolk Herbs are a great, cheap supplier, also good for salvias (another easy to grow, drought resistant Mediterranean flower). There is also a good herb stall in Colombia Road, if you are a Londoner. I like planting salvias, lemon balm or the fruit/chocolate mints (especially strawberry) in places where you brush past them so they release their scent.
www.norfolkherbs.co.uk/plant-type/salvias/page/3/