There's a reason for the height restriction you have, ( a stair behind ) . You have no clearance for a cabinet, and perhaps can use one shelf. : )

Where are you thinking to put a small cabinet? With no water source there it’s more of a quasi bar space. You could maybe put a nice bar service cart on the left?

As Jan said you have room for maybe one shelf…is there any lighting in there at all? Without lighting it will only cast more shadow onto that area. Maybe a single glass shelf if lighting can’t be added.

As others have said, there's not much room. Perhaps there is room for one shelf. If you add that I would make it a glass floating shelf to keep the look light and airy. If you can replace the backsplash please do.

The area is dark and the back tiles and gold colour paint does not help. First I would get rid of the patterned mirror tiles, it does nothing for the space. Since you don’t object to mirror I would add plain mirror to all 3 walls surrounding the counter. Add a floating glass shelf to hold glasses and add a recessed light on the ceiling of the bar. That should make the area look fresh and bright.

Here are some visuals of what I’d consider.

I think you're saying you want to open the dead space above your current bar to add the cabinet or shelves. If that's the case, I think the lighter the better because it's so narrow, and shelves so you don't have to wrestle with doors. Think glass and illumination.

Here are photos of glass and wood shelves. The wood shelves block the light from traveling down to the counter top.

Keeping the counter top clear so you have room to mix or pour drinks will be important and that suggests trying to find more lateral working and storage space. Are you able to widen the bar as well or get storage in the studs on the sides?

When you're reconfiguring and hopefully adding light, an outlet would also be handy if you ever want to use appliances like a blender, coffee maker, countertop ice maker...