That is called the "coping". The tile beyond it is called the decking.
Why do you want to paint the coping? There is no paint that will last long term and when (notice I did not say if) it starts to fail it will look 1000 times worse than before and right now it looks great so........by failing I mean wearing off, flaking, chipping, etc. Then you will have to find a way to remove it all and that will take MANY man hours and $$ to make it happen and even then it will look worse than before
Now you MIGHT be able to stain it and it last for longer but even then it is iffy long term due to the wear.
I know this is not what you wanted to hear but someone has to say it. Let us know the whats and whys of your thinking and we might be able to come up with a different idea what will last long term. There are many creative folks on TFP that love to help people solve problems.
Thanks for answering, it looks quite tired close up we just wanted to brighten it up a bit. There are also a couple of very small cracks as per the picture that I wanted to fill and then was planning on covering the fill with the paint.
Maybe its just a pressure washer clean that's required?
I would start with a good bleach scrubbing followed with a good pressure washing. I bet that brighten it right up! As for the small cracks, I would not worry about them. Anything you do to them will make they stand out more as it will almost impossible to blend them in.
How are you caring for your water? Do you have a pool service or are you relying on Pool Store testing?
That should be easily doable. I would recommend a Xylene based concrete stain. You can get them custom tinted at most paint stores. I did my original pool deck as a concrete surface with a cobblestone pattern and I used a number of different H&C Xylene stains to paint the cobble pattern and it lasted for many years without any noticeable issues. Just patch your cracks and then paint away! Just make sure you prep the existing area first.
Oh I can't wait to see it all clean and pretty after your hard work!!! I bet/hope you will be surprised how nice it looks just with a good bleach scrubbing! Make sure to use regular bleach. None of those pretty smells or splashless as some of it will get in the pool and the additives are not good for the pool. Now regular bleach is very good for your pool.
I am glad they are taking care of you. Will you do me a favor and ask them what test kit they use when the come out please. Also ask them for their latest test results.
Do you know how they are adding chlorine to your pool?