There are two different map layouts for Gaia. One has the capture points spread out over a large area, and the other has them clustered pretty close together. The latter map is much easier to solo; if you get the former map, just restart the mission.
Basically, keep capturing and recapturing the points. The grineer will try to take back some right away, but sometimes they'll leave some points alone for a while. Try to stop any that you see running for a console, but don't sit and play defense in one spot for very long if you aren't in control of all the points.
I don't know the current state of Radial Blind on the Xbone, but it can be a good AoE CC ability. Try it, see how well it works for you, maybe bring along some energy restore consumables if you need them.
I wish you luck. Interception missions are a real PITA to solo. Certain frame builds can help a lot, like Nyx as Starlight mentioned. If there's anything like a recruiting channel on the consoles, try using that to find another player who's willing to help (or who also needs to unlock the node).