If the fort's not touching the sides, then it has to be connected at the top somewhere. It's not currently possible to build constructions downward, nor to drop natural obsidian into a space without it falling out the bottom or landing on something, and so be touching the sides.
You may be able to keep to your challenge by turning temperature off so dwarfs don't burn in magma. Then I guess it might be possible. You dig a stairway down the side of the magma pipe, start digging your volcano fort from the bottom up, using a scaffolding to get out to its lowest point, remove the scaffolding when it's done (make sure the fort is connected elsewhere), pump or drain the magma out of the fort itself, and finally turn the temperature back on. That may be possible, but I couldn't say for sure. The dwarfs may still not be able to survive in the magma even without temperature.
[A cool idea similar to aquifer breaching]
Magma flows just as fast as water. This is not impossible without turning off temperature, but you'll need some luck and extra dwarfs to pull it off. However, to keep building lower, you'll have no choice but to drain each successive level in that fashion. Once one level is cleared, you move down and start pumping from the next level. What you'll end up with is ... well, a drained magma pipe. So, better to just tap a hole into the side at the bottom and drain the entire thing out completely, build the fort, then let the pipe fill back up.