How to install the new one without damaging it?
If using a tool at all, a plastic credit/id/rewards card (use the corner) is the best bet. Tools (such as steel ones) that are harder than the (probably brass) o-ring groove/seat can scratch that surface and cause a leak.
In most cases you should not need a tool at all. Clean the outer surface of the o-ring with rubbing alcohol and a paper towel to remove any grease on the surface, so you can get a grip on it. Pinch it between your thumb and a finger (on opposite sides of the central metal part), and then move your pinching thumb and finger to one side - this should raise a loop of o-ring out of the groove on that side, which you can grab. You are stretching the other side and pushing up slack material on the side you are moving towards.
The latter method has the advantage of not damaging the o-ring, either - which does not matter when you are replacing it, but does matter when you would prefer to re-use it.
Clean the whole metal thing above and including the groove very carefully, then lightly lubricate the new o-ring and check that there is no dirt/sand/hair adhering to it, and roll/slide it down into the groove. Clean the inside of the part that goes over this as well, before reinstalling it.
That looks very much like a rubber o-ring that has been flattened out by the swiveling action of your faucet spout.
Use a thin awl, ice pick, or hooked "dental tool" to remove it. You could also use a very small thin flat head screwdriver. Pry up and over the lip, you want to try to roll it up and off without breaking it. Spray lube might facilitate easier removal.
Take it to your local plumbing shop or "mom and pop" hardware store, they will have an assortment of replacement o-rings to choose from. A knowledgeable sales clerk can probably help assess for proper size as yours has been squished for years. Pick up a tin or tube of waterproof plumbing grease and apply a thin layer to the o-ring and surrounding surfaces that rub.