Since a bike shed be useful for you, I would stick with either treated wood, or composite. Treated is much cheaper, but needs maintenance, where as composite does not. Construction costs are also cheaper with wood as compared to concrete or brick.

I would do it in plain metal. Once and right.

Thanks. Didnt know that existed. will look them up. Any names or something you could direct me to?

Someone mentioned the possibility of using a metal frame, and then use wooden/composite steps and decking. Does that make sense or would it make the whole thing more complicated/expensive?

Pressure treated framing with composite decking treads. Buy a rail kit made of composite materials. Metal and masonry materials and installation multiple times the cost. You will need side and center stringers made of pt lumber.

If you're speaking of the metal stairs, pv, there's more info if you click on the link. Your staircase is a little complex, so this option would be expensive. If money is a factor, best to have the current setup replicated in pressure treated wood and stained. Despite its current state, it is an appears well crafted.
My pressure treated double decks have withstood 20 New England winters and are still solid, if that helps your decision. If you could possibly add a roof, it will help tremendously.

I would get a professional's assessment of the basic stair structure to see if it is ok, and if some maintenance (checking screws, support pieces, condition of wood at the foundation supports, treating wood for protection) will prolong its life. Oftentimes it is the horizontal parts that go first, as they suffer more impact from standing water, as can the bases of the supports. If possible to go with the existing basic structure, replace the horizontal members/handrail with composite material in complementary color. Rebuilding--TMK's post above is the ticket. You should be able to enclose that bit under the highest landing for a bike locker...and maybe even do something a little decorative, as it leads the way to your flat.