If you look for compost which says it has John Innes - that will be the right type. It's really just a formula, and means that the compost is loam based, and has more substance to it. Better for long term planting of shrubs or climbers when they're in a confined space like a container.
If it's already flowered, then it's done it's thing for this year, although if B'sJ is a group 2 clem, it may have a second flush of flowers later in the year. It's not one I've grown so I'm not sure, although it should say on the label. If that's the case, keep up with your watering and a bit of feeding, and just wait and see. You can try repotting it into fresh compost in late winter/early spring to give it a good start for next year,'s display, then each year after that, remove a few inches from the top and replace with fresh stuff, along with some slow release food and plenty of water.
It certainly looks absolutely fine and healthy from the pic, so I don't think you're doing too much wrong!