Update 3: Moments was the first service is stopped to stabilise my system. I already did so about 6 month ago. At the time it was only a hunch but now that the system is working normally i can take some proper measurements:

The display is ordered by memory usage and as you can see Moments right at the top with the highest memory usage and 2nd highest CPU usage.

Moments is using ¼ of the CPU and on a 1GB system ¼ of the system memory. It's clearly not feasible to use Moments on a 1GB system. However with 8GB of memory Moments runs like a charm.

The other thing to notice the Universal Search / System Database combo. A combo because the Universal Search index is stored inside the System Database.

Now that the system is stable I restarted the index and Universal Search has the highest CPU usage and the System Database the 2nd highest memory usage.

Just like Moments the Universal Search / System Database combo would be using ¼ of CPU and system memory on a 1GB system. I would say that using Universal Search is unfeasible on a 1GB system but it's not possible to stop or deinstall Universal Search. All you can do is defer indexing indefinably.

Conclusion: Even with Moments and Universal Search eating up ½ the available CPU the system is running smoothly. I'm looking forward on how fast the system will be when Moments and Universal Search finished there background tasks. Which will probably take a few days.