once per week

Can there be an advancement in rings training, if I do it once per week? 

I have built conditioning throughout the years but I feel like I'm resetting and going back all over again if I do it once per week.

As long as you're training GST on other days of the week, I see no reason why you wouldn't make progress.  It will be slow, because you're not getting constant practice, but if you're already training similar movement patterns or strength requirements than a single rings training session a weeks is more about learning to adapt your current strength to a different piece of equipment.

What about 1 day GST and 1 day rings?

I don't think you would make much progress on a 2 day a week schedule, regardless of what it is that you're training.

My take is that 2days/week can work fine if you're hitting the heavy weights, but rings is just an inappropriate choice for this schedule. To work at the needed intensity on your muscles to progress, you'd be working far too close to your limit for joints and soft tissue to handle for any length of time. And if you were doing less, then you'd be losing ground by not training more often.

/2c