I'd say something about 20 gallon would work. Use a good soil-less media like ProMix.

I think it would depend on how large you want the bush to grow. My brother kept a black currant in a 3 gal pot which bore fruit for several years. He decided he didn't like the fruit and I planted it last winter. I got no fruit the first year it was in the ground.

I'd say you would probably want to go a bit bigger, as I don't think the plant ever got that big and roots don't do that well in a small pot for a long time. I could see growing currants in a 5-10 gallon pot (or fabric container), and probably will at some point.

I think you could grow currants successfully in containers as long as you research your varieties first. I grow a redcurrant (no longer know its name) which grows to 6 feet and spreads even wider. But there are much more compact cultivars. Also if space is a problem, red and white currants can be grown in a very small area in the ground as cordons. Black currants don't take to cordoning however.

Currant should grow well in a container, however you mention keeping it on a paved area. In the summer as the pavement heats up, the currants can easily die from too much heat.
I plant my currants on the east side of the house and they done well with the limited sun.

Bass