How to take care of fully grown orange tree?

I've got a few that look this old.

You have a lot of sooty mould on your leaves, check for citrus whitefly and /or other bugs, and treat those first. You can get stuff that you spray on to help remove the mould as well, but get rid of the bugs first. (also check trunk/branches for borer holes)

Then if you don't have a lot of leaf growth going on, fert reasonably heavy with nitrogen early spring as this will help with leaf growth, doing this though might stop/slow fruit production for a season but in the long run it will be better for it.

After it's got some decent leaf growth move on to a standard feeding regime and keep an eye out for leaf discolouration.

At worst it may be just old and giving up, most citrus have a 'shelf-life' it all depends on the variety.

Edit: just to add, you might have an iron or magnesium deficiency but work on the bugs and leaf growth first, maybe use a micro nutrient spray until the tree is looking a bit healthier, then you can deal with specifics.

I'm not an expert, but I have a couple of 80-90 year old trees I care for. I have had success with cutting out as much dead wood as possible to free up space for new growth. I also fertilize my trees with a natural fertilizer made by E.B. Stone and/or cow manure. The fertilizer helps to ensure the soil is acidic enough for the tree to take up the nutrients it needs and the manure provides beneficial microbes for the soil along with carbon and nitrogen. I apply it by digging a shallow trench on the ground around the tree that is under the edge of the canopy. This is also where my drip line emitters are, so the water can carry those nutrients down to the root zone. I fertilize once in the early spring then once later in the late spring/early summer. The black stuff on the leaves looks like some kind of fungus or mildew. You could try a copper based fungicide (many of them are certified for organic use if that is important to you). Follow the manufacturer instructions, but since citrus are evergreen, it will need to be applied throughout the year for different stages of leaf development.

There's a sizeable orange tree in my backyard and the whole area is unfertilized and no pesticides are used. Some of the leaves are blackened and there are more sticks than leaves. How can I "restore" it?

It looks like heavy infestation from some sort of bug. I think I see scale insects on the leaves. The mold is usually a sign of bugs eating sap and excreting honeydew. It needs to be treated asap. There's lots of tree sprays organic or not, personally I'd use a pesticide that lists citrus on the label. Follow the instructions and spray every surface, including the top and bottom of the leaves. It might need additional treatments depending on what product you choose. After that it might need fertilizer since citrus are heavy feeders. It's possible your tree is reaching the end of its productive life but I wouldn't give up on it just yet.

Prune every 2 year... Give NPK every month... Add rusted nails to the soil where your plant is grown.. Give eggshell powder.. Thats it