Restoring a wooden box with metal inlay
Let me start with a disclaimer.  If it was my box, I wouldn't try to restore it.  There's a good chance of turning it into a mess.  Right now, it shows the character of age.  Trying to restore it could leave you with something very unattractive.
The wood seems bare, but if it had any kind of sealer (possibly including the stain), you will end up with a blotchy finish if you don't sand it first.  It would be tough to sand it without damaging the sheet metal, which looks like it isn't removable without damaging it.
If the sheet metal has any kind of protective finish, the results will be blotchy.  If it has a copper patina, it's likely solid copper sheet.  If there are irregular small blotches of dark patina that aren't consistent with natural sources like finger prints, that would suggest that there might be a protective finish (or it could have been something in contact with the metal at those spots).  The proper way to restore it would be to first strip any potential finish or coating, but it isn't very practical to do that without removing the metal.  So you will be taking a chance with any kind of cleaning.
There's a potential for creating a mess on the wood from what you do on the metal, and on the metal from what you do on the wood.  It's risky to do this kind of stuff after the fact.
So I would advise against trying to restore it, but I'll provide some tips and ideas on doing it if you're set on doing it.

Remove parts that can be easily removed without damage so you can access the surface of the box unobstructed and work on the removed parts without messing up the box.

Do meticulous masking.  Seal off the metal when you work on the wood, and the wood when you work on the metal.  Regular painters tape isn't good enough.  I would put down an edge strip of something like rubber cement to seal the surface, then something like FrogTape, which blocks water-based materials from getting past the edge.

If the filigree had been a surface decoration, you'd be out of luck.  Since it's the edge of embedded metal, it theoretically would allow you to re-stain the wood.  But it won't be as simple as slapping on some stain and wiping it off.  The stain will soil the metal, and cleaning it off the metal will mess up the stain on the surrounding wood.
You can try to apply stain and wipe it off while it's still wet, and see how cleanly it comes off the metal.  You may need to employ a procedure like using a penetrating stain, letting it soak in deeply and dry, then lightly sanding the surface with extremely fine grit sandpaper.  You could also try using a tiny brush to apply the stain around the filigree, being careful to avoid the metal (a very slow process).

Cleaning copper isn't the problem.  The problem is keeping it that way.  Copper starts to oxidize as soon as you stop cleaning it.  Cleaning dissolves the oxidation, and leaves a microscopically rough surface with a lot more surface area.  So every time you clean it, it oxidizes faster.  And every time you clean it, you remove some of the copper.

Many weak acids will remove copper oxidation.  Lemon juice, orange juice, or vinegar are popular.  People use tomato sauce or ketchup (the tomatoes are a little acidic; ketchup also contains a little vinegar).  Commercial cleaners often use citric acid.  For heavy oxidation, people add salt, or use a baking soda paste as a mild abrasive.  You can mix the acid with a little flour to make a paste that you can apply and let it sit for awhile.  There's an interesting comparison of the effectiveness of various popular mixtures: Compare Home Made Copper Cleaners.
Commercial products may clean copper faster, but that's not necessarily a good thing.  They do it by using stronger, more aggressive acid, which will lead to a need to re-clean sooner.  Some also contain a chemical used to purposely create patina.  I'm guessing that's in there to create the need to use the product more often.

If you don't protect the metal, it probably doesn't make sense to clean it.  You can coat it with oil (which will also be good for the wood), so you can do the whole box.  If you use a permanent finish, it will need to be something impervious to air and moisture.  If any air or moisture gets through, the copper will oxidize under the finish, and the finish will make it all that much more difficult to clean it again.

You don't necessarily need to restore everything to it's original appearance.  You can do things to merely improve how it looks now.  For example, touch up the most worn areas of the wood to be consistent with the rest, or restore only the aspect that bothers you the most or is the least risky.  It may improve things just to clean the box.
Sometimes, it's sufficient to just make things more uniform.  For example, the wood has some dark blotches.  Rather than staining all the wood, you may be able to lighten the blotches.